Translation:Can I have your name and your phone number please?
It's kind of implied that "per favore" sounds like asking a favour of someone, when asking for someone's phone number is not asking a favour so you use "per piacere" as a polite tone. In my opinion per favore is also used in more of a formal situations tone, but I could well be wrong here.
As a child, when I asked "can I.....", my father used to reply "you can, but may you?" In English "can" and "may" are not really interchangeable, "can" is to be able to, while "may" is to have permission to. Is there a similar distinction in Italian? "Potere" seems to cover both.
I found a website that says potere is closer to "may" than "can" in Italian and there are times when you would say "can" in English but it would be wrong to use "potere" in Italian. The website uses the example, "I can swim" it says "posso nuotare" is wrong and you should use sapere, "so nuotare." It looks like Italian doesn't have a "can\may" distinction but has a "potere\sapere" one. If anyone has better information please add it. I'm just a student.
Might a native speaker offer a context in which this sentence might occur? Here's my guess: In a large group of people belonging to the same organization? Union? Schoo? In the German- or French-speaking worlds, one normally does not use du/tu with an adult whose name one does not know. The tone here seems polite, so the use of the informal pronominal forms strikes me as odd.