$$P+\rho gh+\frac{\rho v^2}{2} = C$$
Does the $P$ here stand for the hydrostatic pressure exerted by the fluid on its surroundings or the pressure that’s exerted on a given portion of the fluid? I assumed it was the former but Torricelli’s Theorem made me think that I’m probably wrong.
A similar question has already been asked here. So basically P is the pressure energy. You can further view this answer What is Pressure Energy?
Pressure is a concept connected (but not coinciding) with the force exerted on a surface (not on a portion of fluid, which would be a volume).
In the formula you are referring to, $P$ stands for the local pressure in a point at height $h$ and where the local speed of the fluid is $v$. Calling it hydrostatic looks like a misname (since the fluid is moving), but the reason is that it is customary to call "dynamical pressure" the term $\rho v^2/2$.
I do not understand why you think that Torricelli's Theorem would prevent the identification.
