Here in Italy it is not a common name, I know only one person named Santo and to me it sounds a little strange, but you can get used to it. It helps that actually you rarely use the word "Santo" near a saint's name, it's more common using "san" (san Michele, san Pietro). Also I want to underline that in Italian Santo is not connected to Santa Claus at all. Santa Claus is Babbo Natale, so totally different; also Santa Claus derives from Saint Nikolaus, not a Saint named Claus.
I think Santa Claus should be the name of Mrs Claus and his name should be Santo Claus because it's masculine and Santa is feminine. The person who named Santa obviously wasn't all that bright. And they're Italian.
Also I want to underline that in Italian Santo is not connected to Santa Claus at all. Santa Claus is Babbo Natale, so totally different; also Santa Claus derives from Saint Nikolaus, not a Saint named Claus.