Is photoelectric effect a surface phenomenon?

It is somewhat matter of what precisely one would refer to as photoelectric effect.

As far as the radiation-electron mechanism of transfer of energy, there is no direct role played by surface. However, referring to the Einsten's formula; $$ h f = \Phi + K, $$ where $K$ is the maximum kinetic energy of the photoelectron, $f$ the frequency of the incoming radiation, and $\Phi$ the work-function of the metal, it is true that the latter term is depending on the surface and its detailed structure, presence of impurities and so on. In this sense, there is a clear surface effect.

I would say that photoemission is not a surface effect, not normally. Your understanding is correct.

That said, there is also "surface photoemission" due to the $\vec{p} \cdot \vec{A}$ term in the Hamiltonian and symmetry breaking at the surface. As an example, see this paper about silver.