"that's my thing" is a phrase that usually means "that's what i'm known for" or "that's my specialty" or "that's an interest of mine". I make the best pasta, that's my thing. I play soccer every week, that's my thing. I can tell you all about history, that's my thing. So when he says "I'm a cool dad, that's my thing" It means people know he's a cool dad, he loves being a dad, being a cool dad is easy for him. -- If someone says "my thing is", it usually has a negative meaning - it's used to express a doubt or problem with a situation, usually if there's one specific thing that stops you from doing something. This is much less common than the use above, but I know a few people who say it a lot! He never follows through on his promises. My thing is, if you say you're going to do something, you should do it. It seems like a great opportunity, but my thing is they don't offer flexible hours.

"that's my thing" Meaning 'I enjoy doing or being that' and 'it is important to me to be that or do that' An enthusiastic soccer fan could say, 'soccer is my thing'.