It's not as uncommon as you might think.
All three of my kids did the exact same thing. Like others here, I was always wondering if there was something I could do to correct it. The IP program forced them to do T-pushes with both feet, which they would do for the drill, but during free skating, or game play, they always reverted to the same technique. This year my six-year-old has given it up.
They all out grew it without intervention. As a coach, I would be on the ice with them and always reminding them, "use both feet". But however, what I noticed with my kids, and some (not all) other kids who do same thing, is that they were MUCH more likely to favour one side, when they were skating with two hands on the stick. The reason? They were using it as a crutch. Once I recognized this, I encouraged them to skate with only one hand on the stick (unless they were stick-handling, etc). At least they were more likely to use both feet in those circumstances... The other circumstance I noticed that they would revert to one-legged pushing was either during or just after, what from their perspective might have been considered a dangerous or clustered/jammed play... i.e. when they were unsure they were going to be able to remain upright. So I think it might be a defence mechanism for some young skaters, that they return to safety "on their terms", not unlike some beginner players that intentionally fall down if they sense an impending collision with another player or the boards.
I don't know if anything I did, or didn't do, really helped or not with my kids, but maybe watch your player to see if any of the above is the case.
cheers