As others have said, it can't mean that unless a very specific context makes it clear. Why don't you post the clip so we can see the context Maybe then we can be more helpful.

No, that's very strange. It must be something very specific to that particular situation.

What's your story? = What's going on with you?

Hello Hunji. To directly answer, "What's your store?" can not mean "What's your name?". I partially agree with Mag's response, but it WOULD be common to ask 'What's your store?' in an introduction, if you are at a trade show, or a watching e-commerce videos. Example: "Hi, I'm Jonathan, I run a sporting retail outlet... what's your store?" - in that context would be asking what specific area of a sales market someone is participating in.

Do you mean: What's your story? A store is where you buy stuff, it has nothing to do with self introduction.