How does the level of a user rank up? Does it just increase with their starter sylesti or on it's own over time?

Level user is determined by the highest level of pet you've levelled. Doesn't need to be your starter but I expect, early on, it would be people's starters that would be a higher level.

By way of example, just because you buy a level 60 pet doesn't mean you'd be level 60. You'd increase to level 60 if you had a level 59 pet and the pet reached level 60.