You do need to ascertain the facts. Ask him a straight question - what exactly has he said about you, to whom, and were you named or otherwise identified (SIL, family member). Insist on a straight answer before you allow the conversation to develop, at all.
If you have been named or otherwise identified, to people who could possibly know you - whether or not a health issue was referred to - I would be absolutely fucking livid, in your situation. I would make VERY clear to him that in future he WILL respect my privacy and my right to a private life.
(You could, if true, point out that anxiety, or just uncertainty and social discomfort, caused by knowing that others know more about you than you'd like and may be wondering, or asking you, what the problem is, is likely to make your MH problems worse not better).
Crucially, he is not your vicar, you are not part of his church community, or even of the wider church family.
I think it's normal if you're part of a church community to be prayed for in hard times 'our friend x, our sister y, at this difficult time' etc. But that's ok because it is a social and spiritual community you've opted to become part of and share yourself with, to an extent.
If he's hard to talk to, interrupts and tells you why he's right, instead of listening actively and giving straight answers, I'd set out your concern in a brief, polite letter to him.
If he odd share any info about your specific condition I'd be writing a letter of complaint to his manager. But, he's your BIL, so that's not necessarily what you should do.
If he wanted to pray for MIL and her difficulty in coping with difficult times, and she is up for that, then fine, if it's kept vague.
Personally I would feel exactly the same if it was diabetes, or a bereavement, rather than MH because the point is that my private life may be my family's business by default but it is not his congregation's business.
If he was well-intentioned (rather than self-serving, which is quite possible) then he acted stupidly and needs to reflect.