I have been under the impression that the definition of enthalpy as
$$H = U + PV$$
is purely mathematical, the result of taking the Legendre transform of the internal energy $U$, and merely provides a convenient expression to find the differential form of the enthalpy,
$$ dH = dU +d(PV) = TdS + VdP + \sum_i\mu_i dN_i$$
which is a useful expression for performing experiments at constant pressure, which I take to be the main reason for even defining the enthalpy in the first place.
Because the natural variable $P$ in the enthalpy is intensive, it cannot be integrated using an Euler integral. But in several places I have seen the expression
\begin{align*} H &= U + PV \\ &= TS - PV +\sum_i\mu_i N_i + PV \\ &= TS + \sum_i\mu_i N_i \end{align*}
defined as the Euler integral of the enthalpy. It seems we must have
$$ dH = dU + d(PV) = d(TS) + \sum_i d(\mu_iN_i)$$
But I do not immediately see why
$$ TdS + VdP + \sum_i\mu_i dN_i = d(TS) + \sum_i d(\mu_iN_i)$$
If I expand the differentials, will I eventually be able to use some Maxwell relation to arrive at an identity? Is there a meaningful physical interpretation of the term
$$ U + PV $$
or is it, as I suspect, purely mathematical?
