The answer no one wants to hear, that I hate to give, and which implies the privilege of easy job motility, which isn't as realistic for everyone as some people like to make it sound:
Start looking for an exit strategy
...if you don't like what's going on, because you have a ton of warning signs that more directly and strenuously pushing back on it isn't going to end well.
Also, he thinks that because there was a death in my family, that I will move away and quit the job when I have re-assured him that my husband and I have no intention of tearing our kids out of the local school just to move to keep a relative company.
So, as someone who has dealt with her fair share of sexist crap in this industry, I have to ask if this is an attitude he'd take towards the male members of the team/his reports. Because it seems pretty suspiciously likely to be bias motivated.
I have tried to do that and he complains saying that he feels like he does not know me even after working together for 2+ years and that people might not warm up to me in the future if I don't overshare.
That's some bizarre bullshit. We're at jobs to work, not to have everyone else in our business. It's perfectly possible to be very surface level friendly and have a nice time and great professional rapport with everyone else on a team while sharing basically zero personal details.
If you really want to explore avenues of trying to deflect this first
... what is something that's not actually personal to you in terms of how you feel about it, but can sound personal to an outside perspective? Something like a hobby that you don't feel overly intimate about, etc. Sports teams or athletes you follow, if you're into that. Some algo or pattern you came across and find interesting. New games/big name movies that have come out that you're curious about (but not necessarily ones you play/watch). "I heard about this and am curious" is always a great way to get other people talking if they are interested, without having actually shared anything about yourself (although related questions ARE likely to relevantly occur then).
Equally, there's just getting into talking on something even if it's something that you think literally no one else will want to hear about. It can honestly be better if it IS something that no one else will care to listen about, if you don't actually intend to share anything that might be relevant about yourself. If your team meetings devolve into chit-chat at the end, consider talking about it randomly when there's moments where that would make sense. There, you've shared something "personal".
But either way, you don't have to make close personal friends to have work friends who know next to nothing more deeply personal about you from outside of work, but are still definitely a form of friend and quite amicable.
Digging on this by a supervisor has a lot of nasty warning signs, it's not really appropriate with the related power dynamic, and his posturing around his doing it only makes those warnings much stronger. Sure it could be "innocent" other than disgustingly patronizing on his part. But it could also not be, and regardless it's honestly inappropriate behavior.
You don't mention yourself feeling anything in that direction, so maybe it's nothing. But reading what you've relayed has discomforting aspects to read. Maybe I'm just over sensitive from being tired of dealing with related kinds of crap when I just want to work, myself.
Do you know what's going on with the rest of your team and your manager?
Is the frequency of your 1 on 1 meetings the same as the rest of the team? If you feel up to it, can you feel out some of the other people on your team on how your manager interacts with them, and whether he prompts them for personal details?
Where's HR on this?
Do you even have real HR wherever you are working? (having, myself, worked at one place where "HR" was the marketing director's girlfriend, hired for the position literally because of their relationship, and she had no authority while equally maintaining exactly zero privacy for anything divulged to her)
Before you consider things like escalating "my supervisor is being creepy personal" to HR, even if they are even at all decent, one solid strategy that's open to you is that you should NOT be telling your direct manager these things if they make you uncomfortable. Tell your HR rep. First. Before talking to your manager (be aware this may turn into him expressing hurt that you "went around him", even if it's an entirely appropriate way to handle these types of things in many places). Then tell your manager that you've already communicated the details with HR so it's all taken care of. If pressured, simply say they're personal and you're not comfortable discussing them further.
Over-emphasize your discomfort. Are you capable of guilt tripping him without actually divulging details? Whatever it is that you're not sharing with him, keep it very vague but over emphasize how badly impacted you are by it and how it's a terrible thing but equally imply it's not something that would be socially appropriate to be asking someone to share, much less to heavily pry on. For example, with the death in the family, saying something like you just need the space to grieve and can't deal with talking about whoever died, and acting shocked that he's now talking about you leaving and other things when you're focused on just coping with your grief and the healing process and he's just making it harder for you by not letting you have your space you need for that, when things like leaving are the farthest thing from your mind. Or whatever fits your personality.
Regardless, if he still presses on wanting details even simply at the point where you've said you already went to HR and you're not comfortable discussing it more, you really do need to consider whether taking his behavior to HR is going to end at all well for you, and I'd strongly recommend having an alternate exit ready even if you DO think it will. At which point, you may as well take other opportunities if they appear. Fighting a potentially bias related dismissal sucks, and even if you have a union representing you and end up either in arbitration or with an adequate settlement offer, you're unlikely to end up with anything actually truly adequate to the stress and professional derailment, compared to just getting out ahead of it blowing up. Especially if it's something where it's not immediately obvious/empirical, but is simply some behavior that "happens" to fall into related bias avenues and you can't seem to get to shut down.
Either way, keep track of these conversations with him. Take notes during your 1 on 1 meetings, if you don't already. As in, take notes not just of the relevant work things, but of the patterns of behavior and specific incidents related to your manager. Get in the habit of carrying a notepad or note taking device and using it, if you're not already. There are definitely worse habits to have, so something positive at least can come of it all!
Again, maybe he does this to all of the team. But maybe he doesn't. And regardless, you shouldn't have to do your manager's job for him, without having his position or a comparable one to doing that kind of work. It's one thing if doing the work will get you the position or some similar upward/forward motility. It's another if he's just going to mooch off of your work and then claim credit to whoever manages him.
You certainly could, hypothetically, go up the chain with "wow, did you know the person supervising me is really crap at his job and keeps digging into my personal life and pressuring me to share more about it?" but... what would you expect to get out of that? Other than a suddenly bad performance review and put on a PIP that's suspiciously lacking tangible review measures, before being told you haven't made adequate progress?
It's horribly unfair to be stuck in this position, with these options. Who knows, maybe simply taking a firm stance with him on "sorry but that's too personal and I'm not comfortable discussing it" and shifting the conversation will work. But it sounds like you've already tried pressing him to back off and he hasn't. Maybe you weren't direct enough? But it kind of sounds like you were, given what you've said of his responses on how he "does not know" you.
When you're at work, it's hard dealing with coworkers who can't respect your boundaries. It can be nearly impossible to have any clear path to dealing with it when it's a manager and the level of impropriety can still be claimed to be innocent, because if it is in any way malicious or coming from a problematic personality/behavioral aspect, trying to more seriously push back can easily result in effectively retaliatory circumstances with highly negative professional repercussions and little means of proving that they stem from that.