I was recently thinking about Finnish in the light of noun classes, or rather its alleged lack of them. It's well known (along with other Uralic languages) for not having grammatical gender. However, there are many nouns that share various systematic stem changes when affixed. One such group of nouns, the "-nen words", switch from -nen to -s- when an affix is attached:

Suomalainen
Finn
'Finn.'
Suomalaiset
Finn-PL
'Finns.'

Another such stem change from /Vs/ to /Vks/ occurs in other words:

kysymys
Question
'Question.'
kysymykset
Question-PL
'Questions.'

The words that fall into these classes do so based on their phonemic structure (e.g. ending in /nen/) rather than some semantic similarity, but plenty of languages have pretty arbitrary membership for their noun classes too.

Could/are stem changes such as the ones in Finnish considered a type of noun class?

That sounds like a "declension class" to me (look into traditional Latin grammar regarding that), rather than a [semantic/grammatical] "noun class" or gender system.

Generally speaking gender/class systems do include some kind of agreement. They ALSO happen to pattern like you're describing. Look at Swahili as an example of something just like what you described, except that it also then has agreement with adjectives and verbs following similar morphological patterns.

Classes are arbitrary but not entirely random or phonological. Some words may belong to one just because they have a similar sound or for no particular reason at all, but overall the system will show some semantic patterns.