While "any fish" is the object in the English sentence, "kalaa" is actually the subject in the Finnish sentence. The typical structure of an ownership clause in Finnish is adverbial-verb-subject, where the adverbial is in adessive case and expresses the owner, and the subject is in nominative or partitive case and expresses the owned thing. The reason why the owner is expressed with an adverbial is the fact that adessive is a locative case and therefore expresses location.
The partitive gets used for many things and it's rather tricky for language learners, both old and new. For some words the partitive suffix is -ä or even -ta or -tä depending on the structure of the word and the vowels in it.
If you wanted to ask "Do we still have the fish?", you'd say "Onko kala vielä meillä?".
"Saisinko kalaa?" - "Could I have/get some fish?"
"Saisinko kalan?" - "Could I have/get a (one) fish?"
'Onko meillä vielä kalaa' can be translated a few ways. "Do we still have (some) fish?" "Do we have (some/any) fish left?"
In these translations vielä is either 'still' or 'left'. 'Some' can be included or omitted before a mass noun. Likewise, 'any' can be included or omitted before a mass noun in many questions.