You can see your current level on the Profile Menu of the Watch Menu. Players can currently reach up to level 50 in Rec Room. Your progress towards the next level is visualized by the closing line around the level number in the Profile Menu. (Note that this closing line is updated only when you start Rec Room.)
You need level 15 to be able to gift Cosmetic Items to other players.
Rec Royale has its own leveling system (called "tiers").