BYU has two basketball NCs. They just don't have any NCAA tournament championships. The years BYU won it's two NIT tournaments they chose to go to the NIT over the NCAAs because the NIT was the more prestigious tournament. Basically up till about 1970 a team that won the NIT could claim it as a legitimate national championship. In 1944 when Utah won the NCAA tournament they had already lost in the first round of the NIT - which kind of sullies their claim to an NC in 44. The team they played in that championship game doesn't help either. That being said, Utah like BYU can claim 2 NCs because of their NCAA win and one NIT win in 47.

Prior to the 50s the NAIA champion was also considered a legitimate national champion.

This just emphasizes the fact that despite how football rules the roost now days , BYU's and Utah's athletic traditions are based in basketball.