Your sentence isn’t wrong but it should be noted that in your example, ‘more’ isn’t necessary and would usually be omitted by native speakers. It’s implied by the verb “to add”. For example You can add (a) soup for two dollars. You’d use ‘more’ with a different verb. E.g, Do you want soup for three dollars more?

Yes, they mean the same. In both sentences, "more" acts as an adverb to modify "one". In either case, "more" means "in addition" or "on top of that". If you simply say “Would you like to add soup for one dollar?”, the meaning is exactly the same. If you omit "dollar" and say “Would you like to add soup for one more?”, the meaning is the same so long as it is understood that you are talking about dollars. When you do that, "more" becomes a pronoun.

Both versions of the sentence are grammatically correct, but they have slightly different structures. The original sentence is: "Would you like to add soup for one dollar more?" And your proposed change is: "Would you like to add soup for one more dollar?" Both versions are clear and convey the same meaning. It's a matter of personal preference or style. Choose the one that you find more natural or fits better with the overall tone and style of your writing or conversation.