Hi seventeen,
I think it varies depending on the child.
With DS1 it happened quite early. At 2.3 years old, he was already saying sentences of 5 to 6 words in language A just like your son.
I wasn't quite sure how much of language B he knew as we chose language A as the home language.
He learned language B at nursery (only 2 days a week), and he was exposed to language B whenever we went to a playgroup, out in the street, or if we met friends who didn't speak language A.
Up to that point he had only said about 3 words in language B to me, and I asumed he was using those few words and maybe more at nursery.
Then one day his nursery teacher came to me excited because he had said "sit down" to her for the first time. That was when I realised he wasn't talking at nursery despite being a chatterbox at home and knowing how to say "sit down" in language B for more than 3 months already.
I thought the reason he wasn't confident enough to try to talk in language B was probably the lack of vocabulary.
I decided to give him a gentle push/help, and from then on, when we read picture books and he named the pictures he saw I said to him "Yes, we say this word in language A, and they say this other word in language B at nursery". I also asked DH to read a story to him in language B (language B is DH's first language, although he agreed to use his second language at home to give language A more exposure).
I think this gentle support was all that was needed because in less than a month he started talking a lot in language B and very soon he was also labelled a chatterbox at nursery.
I think that because of this, from very early on he acquired words to label language A and language B. And he was adamant that DH would only read to him in language A, and not in language B. He always showed a strong preference for the minority language.
As soon as I saw language B starting to develop, I stopped referring to it at home, so that language A continued to be used exclusively.
DS1 is now 6 years old, and fully bilingual, never gets them mixed, and picked up language B perfectly fine at school in terms of learning to speak, read, write, etc.
I have another son (DS2) who is only 2.4m now, and with him, I haven't seen that he differentiates language A from language B like you mention.
He doesn't mix the languages. In fact, he has over 200 words in language A, can count to 10, can say sentences of 5 to 6 words, but only knows 3 or 4 words in language B.
He doesn't go to nursery. Only hears language B at playgroups, and when we see family once a week or every two weeks.
He doesn't seem to have labels to differentiate the languages, and sometimes I wonder if he knows/notices that other people speak a different language to us.
I'm quite curious to see how it would be in his case.