Even just these initial thoughts are very well-constructed and helpful.
I'm not at all a linguist, ...
Great! If you take it one step at a time, I think you could make some progress with this, and feel free to ask specific questions as you try one step or another. But to start just gather data and look for some kinds of patterns in it.
Also regarding qualifications, I don't have a background in any kind of this sort of psychological condition. I do, however, analyze lots of diverse languages every day, and you said that you've talked to specialists who don't know what's going on, so I'm looking at it from a purely linguistic perspective to see if that's helpful.
I am a native English speaker. The only other language I have studied is basic Latin, though only in text.
Best guesses, then:
1. This is not Latin, doesn't sound like it (either in pronunciation or the way the sounds combine), although it could potentially be somehow influenced by how you imagine (written) Latin would sound in your head.
2. This is not based on your experience with another language, except possibly in the same sense as Latin your impression of another language you've heard. Plausibly it could be something you heard at a particular time (a foreign-language-speaking friend or family member), or even your impression of someone 'talking in tongues' in a movie. But there's no evidence for that. Regardless, it's not actually any of those things, even if it's your impression of how something (or several things) sound. To me it sounds just a little like Russian, for example, but not beyond a little resemblance. (Specifically the way you say certain consonants, something like
palatalization in Russian, but that could just be related to an intense manner of articulation, so a sort of 'accent' rather than linguistic feature.)
3. All of that means that the by far most likely scenario is that this is based on English in one way or another. Your brain hasn't been wired to be fluent in any other linguistic system so it wouldn't by default bring in things too foreign to English. There may be some differences, sure, but without any additional evidence I'd expect it to be similar rather than different, from a structural perspective.
As for meaning and context, I really have no clue. It's as if I'm listening to another person speak a language that I've been around, but have never learned.
OK, at least that's something to start with.
To look at this abstractly for a minute, something very interesting that Dr. Con Slobodchikoff proposed, related to interpreting animal communication, is that the best (and maybe only effective) way to understand what animals are "saying" is to find evidence of their behavior that changes correlating to certain sounds. We can't anthropomorphize them and assume their intentions and motivations are the same as ours or what we would assume them to be. He's got a very nice short video about that here, which actually also serves as a basic introduction to structural analysis, plus it's just fun to watch the prairie dogs:
https://www.youtube.com/watch?v=y1kXCh496U0Of course your situation is different (and I don't mean to imply anything negative) but from a diagnostic and translation perspective, one difficulty is the same: we don't know why you are saying what you are, and it could be for any reason. If we can't observe any changes in your behavior, we may never be able to translate it, because we just don't have any reference points. We could still analyze it structurally to find out if there does appear to be a structure something like a grammar, but we might never work out what any particular words mean, especially because we don't even know if you're talking about something in particular (and what that would be), or expressing abstract feelings/thoughts, etc.
Even assuming it has roughly the structure of other normal human languages, there have been situations of trying to decode unknown languages that have been very challenging. Most of the time the success depended on finding a shortcut, while a few times some progress has been made from linguistic data alone. Some languages remain untranslated, especially those with very little surviving text.
A well-known example of a language being deciphered after a substantial amount of time is Ancient Egyptian. It was known that the hieroglyphics were language but modern scholars did not know what they said or very much else about the language. What changed this was the discovery of the Rosetta Stone, which had the same text in three different writing systems. The first was Egyptian hieroglyphics, the second was Demotic script which was essentially the same text but represented differently in a (more transparently) phonetic way, followed by Greek. All of these pieces combined allowed not just translation of the meaning and eventually to the meaning of each word, but also to the spelling based on the serendipitous use of proper names in Greek and Egyptian. Of course in your case there's no problem of deciphering a writing system (although making one would be helpful), but the same applies: without that Greek text telling us what is carved on the stone, it's hard to know where to start.
Another situation is Hittite, which was discovered at least a few decades before it was deciphered. It was eventually deciphered in part because it was determined to be related to the Indo-European languages (including English, Latin, Greek, Russian, Hindi, and so on).
Finally, some information can be determined even without that kind of information, like by looking for very common letters (probably vowels, etc.) or very common words (probably serving a common grammatical function), but there are limits.
So moving on from analogies, the point is that without any correlated motivation, behavior or meaning, it's going to be hard to actually translate it.
I'm not at all a linguist, but I have noticed interesting things like, "Bac'sunde etra un dei, Bac'sunde da ei un dei." Whether that's just some feature of my brain coming up with something new to say, or an actual structure of some kind, I don't know.
The repetition is either structurally meaningful (where Bac'sunde is some kind of important word, apparently), or it is just because it sounds good to you.
The null hypothesis (that nothing interesting is going on in this data) would be that you are simply babbling with sounds that are desirable in some way, and that you would repeat sounds that are desirable.
But even if there isn't some deeper meaning behind all of it, from a linguistic perspective it would be fascinating to find out just how much linguistic-like structure has developed, given that you are human and humans naturally develop grammars, at least as children. That's another relevant question: have you been doing this since childhood, since say 5-10 years old? After that you're an adult in the sense of not really being young enough to learn a language as a native speaker, so
potentially if you started doing this before that point, it could have developed like English did, where you are now a fluent native speaker. If this developed during adulthood, then you might be more like a non-fluent second language speaker. The relevant theory is that according to some your brain's natural ability to develop linguistic skills shuts off after the so-called critical period (ranging from about 3 to 12 years, ending by puberty, but most studies show it a little earlier than that, some as early as 5, but it depends on what you measure and how). That's all speculation in this case, of course.
As for what to do next, from a purely linguistic perspective, you should gather more data. Transcribe the recordings you have into a consistent spelling system (since you know some Latin, it would be better to spell it something like that rather than using the messy spelling of English, or you can look up more information about phonetic notation, not that you need anything technical for this really.) And if you can keep recording, that would be good.
My working assumption would be the null hypothesis that this isn't really a full "language", but obviously I don't know, and the question is interesting. Even if not, how many linguistic features does it have?
The only similar thing I can think of is that I've heard anecdotally (I don't know about much research on this) that sometimes twins (or other siblings) develop their own secret language that sounds like babbling to others. I'm not sure if those systems are as expressive as a language like English, but apparently they can understand each other. So it's not impossible for a human to do what you're wondering about.
For now gather/organize data, and see what happens. None of this is psychological/medical advice of course, but as long as it doesn't interfere with anything, go ahead and see if you can figure out some patterns. If you end up gathering more data, in theory I wouldn't even mind taking a look at it and seeing if I could find some relevant patterns, but if I started working on this in detail (and it wouldn't be immediately-- I have a lot of projects at the moment) I'd like to potentially consider publishing the results if they were interesting (anonymously or with credit to/with you, of course, whatever you prefer), but that's up to you. If you don't want to do that (or if I don't end up having time, or just not for a while) I'll be happy to keep discussing it some here on the forum if you want to try it yourself.
I also have to imagine there are some speech pathologists out there who would be able to do something about this. They might not find it to be medically/psychologically relevant what the content of the babbling is, but aside from that I don't see why they couldn't try to figure it out. You'd have to find someone interested and with a background in linguistics, but many undergraduates who study linguistics go on to get a master's degree in speech & hearing science. So some of them are out there and capable.
Be prepared, of course, for there to be no deeper structure. My interest is just that, I don't mean to mislead you in case it turns out to be just babbling.