I think that you should stay away from such people in the future, this communication will not bring anything good.
The phrases she uses are very similar to manipulation, she wants to show that she cares, but on the other hand, it looks like a person keeps you on a short leash.
If I were in a similar situation, I would not let this person come close to me, but keep them at arm's length.
But of course, it's up to you to decide how to behave with her and whether to communicate with her, and I agree with your opinion that it all looks very strange.
Good luck with making the right decision.