It's completely fine to get to know your coworkers. But let the oversharing ones (the extroverts) congegrate together naturally. We had someone like that on the team. Was likeable enough but wouldn't stop talking about everything in their life. That was their choice tho and I am fine to listen but I'm not "sharing back".
And then there were introverts. Some that spoke very very little and some that were in the middle. They randomly shared some things from time to time in quieter moments with less people around and when they felt comfortable. Very real people and it was nice to get to know the a bit more.
I'm more on the introvert side there and if you group pressure me into this sort of thing I will remember that for a very very very long time. As in we are never gonna be friends. I'm still sour about what an extrovert HR person did 4 years ago and I don't speak to them any longer. They are also showing from time to time that they haven't changed and they'd be the type to use whatever came out in an oversharing session against you.
Now tell me to "report them to HR".
Now I understand that with remote work there may not be enough space for chit chat. But there are other ways than a forced group pressure situation to set that up.
For example, we decided as a team when Covid WFH started to basically use our daily 15 min stand-up for whatever. Sometimes we just did stand-up and we're out of there after 5 min coz nobody felt like talking. Sometimes it'd be a half an hour of just talking about whatever. We also have a dedicated time each week blocked off on the calendar for it as well and when we have ad hoc working session calls we get in some chit chat here or there as well. It's not all just business but it can be.
RE cat: see this is the internet and it's an anonymous forum. I used that example because our cat actually just died a few weeks ago. You thought it was over the top. Interesting indeed.
I don't quite get the difference between a blocked off time for chit chat and the way the manager asked for it here but totally, being forced like you have to say something you don't want to sucks.
I guess the difference is the same what I mentioned about stand-up changing: The team decided to do that. It wasn't a manager saying that's how we do things now.
EDIT: I also just realized you gave out condolences. See if I was in the situation from the article I might not be able to quickly come up with something fake and I'd instinctively just mention the cat thing (coz I don't want to be the guy that gets shunned). And then the next few weeks are gonna be very uncomfortable coz people will keep mentioning it.