- It costs a dollar or more.
- It costs a dollar, or it costs more than a dollar.
Do they have the same meaning?
Yes, they convey the same sense. However, the former is better as the repetition of "costs" in the latter is unnecessary.
Yes, they're the same.
"It costs a dollar or more" means exactly one dollar, or an amount more than one dollar. Amount ≥ $1.
"It costs more than a dollar" means amount > $1, but not equal.
Bear in mind that someone may be approximating. If you asked someone the price of a product and he said, "a dollar", I wouldn't be surprised if it was actually 99 cents or $1.05.