Well the name Yamato was an alias given specifically in conjunction with Naruto's training. Considering Yamato's primary role ever since his debut has been training/ subduing Naruto...it makes sense that everyone calls him that.
The thing is, that we don't know for 100% certain whether or not Tenzo is his real name (yes its Tenzo not Tenzou; official Japanese names often have silent u's but, in the Naruto universe at least, its spelled Tenzo). The fact of the matter is that Yamato has spent his entire adult life working in conjunction with Anbu, so it is very likely that Tenzo was the name he used during all his Anbu dealings. The way he got mad at Kakashi for calling him Tenzo seems to imply that it is an alias associated with a dangerous part of his past, but not his actual real name.