He actually looks giddy in this picture.
Also, Steve's wife, Lisa commented on the photo "just when we thought our week couldn't get any better..." so clearly a family of fans.
Yzerman is basically Taylor's hockey idol. He was the captain of the Red Wings for forever. He said his brother was an Yzerman fan growing up so he became one and he's always been an Yzerman fan, still is. He follows both the Red Wings and Tampa Bay (where he is now the GM) because of him.
When his friend, Neil, was in the Wings organization he played a pre-season scrimmage game at the Joe in Detroit and Taylor flew in because he'd be playing with Yzerman...he's told the story a few times about meeting him.