No, carrying capacity applies to objects actually picked up and carried in the literal sense (which could include clothing and armor and things worn) but not one's self.
It's hard (perhaps impossible) to find a direct citation for this, but the carrying capacity rules are written in the plain English sense. However, there is an indirect citation:
From the PHB, emphasis mine:
Carrying Capacity. Your carrying capacity is your Strength score
multiplied by 15. This is the weight (in pounds) that you can carry,
which is high enough that most characters don't usually have to worry
But, the random rules for rolling up height and weight (long enough I don't want to reproduce them, but reproduced and discussed in detail here) tend to put several races near or above their own carrying capacities with strictly average rolls, which means they definitely would be worrying about it.
It's just not possible to reconcile both of those rules, so falling back on the plain English meaning of carry-- pick a separate thing up off the floor and move around with it-- is correct.