Ciao

Ha scritto sopra il suo name

How come it is  “ha not hai” if in the translation it has “You” in brackets. All these idiosyncrisies drive me a little crazy. I guess  English is the same. 

Sebongela

“Hai" is the conjugation for Tu, the informal “you”. 

“Ha” is the conjugation for lui/lei/Lei, which is “he”, “she” or the formal “You”.

In the example you gave, “ha scritto sopra il suo nome”, the “ha” refers to he/she (the person writing his or her name). It could also be translated as “You (formal) wrote above his name”, which might explain the (you). 

The full example that I think you're using is “Ha scritto sopra il Suo nome e i Suoi dati?” 

This is a customer service person politely asking a customer if he wrote his  name and information on the luggage. “Ha” and “il Suo”/"i Suoi" are all used because the person speaking is using polite, formal language.