Do the two sentences share the same meaning?

It doesn't seem that way to me.

"Do you have something on your mind?" is a way of asking if you are thinking about something in particular.

"Do you have something in your mind?" doesn't sound right, but it might make sense in a science fiction movie where telepathic aliens were invading someone's mind.

But that's from a US perspective. I don't know if it is the same in the UK.

"Do you have something on your mind?" asks if the person questioned is preoccupied with something. I can't think of a situation in which one would say "something in your mind," however. One might ask: "Do you have something in mind," meaning, do you have an idea, do you have a plan.

Thank you both.

Yeah.