Is the song name "U and I" supposed to be Ui's name?
I'm not really sure what you mean when you ask whether the song is "supposed to be" Ui's name, but the following things are true: 


The title of the song is the same in Japanese ("U&amp;I").
A Japanese listener would certainly recognize this as the two letters in the romanization of Ui's name.
It is also likely that a Japanese listener would also recognize these as meaning "you" and "I" (as in "me"). (Compare: You wa Shock!)
Hence, I claim that this wordplay is clearly intentional, and that Japanese listeners would recognize this. 
The song is quite explicitly "from" Yui "to" Ui. I don't know whether the dub translates the songs, but in essence, the lyrics are all about Yui saying how she needs Ui to be there for her, and how thankful she is for Ui being in her life. 


Hopefully this answers your question.