I agree, that you should first check the sizing of his skates. Next, yeah, don't tape his ankles. In doing that, you're basically using that as a crutch to hold him up instead of letting him develop the strength in his ankles to do it on his own. In addition ankle mobility is key to getting into a proper stance and developing a proper stride.
Bure, was a grown man with massive leg strength. Pro players are such powerful skaters they can flex the boot and crease it. I don't think a 6 yr old kid has that kind of power. And finally, I've heard speculation that one of the reasons Bure had knee problems was because he taped up his ankles like that. Doing that put more stress on his knees because he couldn't use his ankles as much to absorb/cushion the stresses of being such a powerful skater.