For work-related discussion, there can certainly be an expectation within your job duties for you to engage to a certain degree. Some people may not be capable of reasonably engaging to the expected degree, in which case they may simply not be a good fit for certain roles. Although there is also some possibility to discuss the issue and redefine expectations to accommodate any personality type or neurodivergence that someone might have.
But it sounds like this isn't really about work-related discussion. What I said below may apply to a reasonable degree to work-related discussion as well, although if this is the issue, I'd probably try to discuss the issue with a focus on the job and what yourself and others can do to make sure the job is performed roughly in the way your manager wants.
For discussions of topics outside of work, I wouldn't say the expectation falls within job duties, but it's rather a cultural expectation. This may seem like a negligible distinction for some, as companies typically also expect employees to be good cultural fit and they can and do commonly quite easily get rid of employees that aren't good cultural fits. But depending on where you're at, this may be more difficult for a company (a company may also get into discrimination territory if there's a possibility of neurodivergence). Cultural expectations are a lot more vague, open to interpretation and not as enforceable. Culture could be either more and less able to accommodate different personality types or forms of neurodivergence, because it ultimately comes down to how accommodating your boss and coworkers are, separate from what's needed for the job to get done (and it doesn't sound like your boss in particular is all that accommodating).
As a very introverted person with autism, I'll say you have no ethical obligation to engage with non-work discussion on any level beyond basic politeness (e.g. if someone asks a question, give at least some answer, however brief, which it sounds like you're already doing, and possibly add a "and how about you?" if relevant). It's a failure on their part to not be considerate towards people who are more introverted. Of course, this does assume you didn't misrepresent yourself during the interview (they could choose to not hire introverted people, but if they hire them, it's their failure to then expect such a person to stop being introverted).
That said, they seem to already be placing this expectation on you. If you don't meet that, it could increase conflict and could potentially lead to them firing you (depending on local laws and how important this is to them).
So I can't tell you what to do, but your options would roughly be one of the following:
Keep doing what you're doing and just brush off what your manager said about it (this'll probably have the worst outcome as far as your job is concerned).
Keep doing what you're doing and offer some explanation or push-back to your manager, e.g. point out that you're introverted and/or say that it makes you uncomfortable to discuss your personal life or that not conducive to your mental health to engage in small talk to that degree.
Take some steps towards meeting their expectations. It could just small steps or large steps. You could, for example, just add a sentence or two beyond what you'd typically respond, or using more emotive/positive words (like "good", "fun" or "great", instead of "fine"). If asked "how was your weekend", rather than just saying "it was fine", you could say "it was great, I played some games" or whatever.
Some combined approach, where you offer some explanation but also try to do more.
If this is really too much to deal with for you, other options would include quitting or taking some vacation or sick leave (how viable any of these are depends on your contract and local laws). Although I'd probably first suggest therapy (which can also help with any of the other options), as therapists can help you deal with anxiety and help you deal with and resolve interpersonal conflicts.
I'd personally have much more of a problem with someone who makes a habit of trying to answer questions for me, especially if they do so incorrectly.
I'd say you'd be perfectly justified in calling that out, whether in public or in private. But calling out the behaviour of one's manager often tends to not work out that well. So what you do with this information is up to you.
It seems relevant to point out that it helps to use "I" statements instead of "you" statements when bringing up issues, e.g. someone is more likely to be receptive to "It makes me uncomfortable to discuss my personal life" or "I am uncomfortable with discussing my personal life", whereas it comes across more confrontational and blaming to say "You make me uncomfortable by asking me about my personal life" ("it makes me uncomfortable for you to ask me..." is probably also not great).
