Gibb's phase rule is much simpler than Euler's formula.
First of all, we're talking about a phase diagram which has $C + 2$ dimensions, where $C$ of the coordinates are chemical potentials for the $C$ many components, and the extra two are pressure and temperature, as hyportnex says.
Suppose there is a special submanifold of the phase diagram which lies at a co-existence junction between $P$ different phases. Then Gibb's phase rule just says that since $P$ is the codimension, the dimension of this submanifold, namely the number of degrees of freedom $F$, is the dimension minus the codimension, or
$$F = C + 2 -P.$$
It's equivalent to the rank-nullity theorem in linear algebra. This is a far simpler result compared to Euler's formula, which is a global property of graphs on the 2d sphere. Gibb's phase rule meanwhile is simply a local property in the phase diagram, which has different meaning at different points. It tells us nothing of the global structure.
If one were to invent a correspondence between them, it would be quite ad-hoc, since such a correspondence would have to say that there is something very special about temperature and pressure, ie. about the number 2. In topology 2 is very special, since it is the largest Euler characteristic among connected surfaces. If we included another tuning parameter, then the phase rule would read
$$F = C+3 - P,$$
and there is no connected surface of Euler characteristic 3, so nowhere to draw the graph that's supposed to be associated to each point of the phase diagram.