Let F be a totally real field. We know that we can define a Hecke operator $T_\mathfrak{m}$ on the space of Hilbert modular forms over $F$, say with some level structure, for any ideal $\mathfrak{m}$ that is coprime to the level.
Let's say we have two eigenforms $f_1$ and $f_2$ whose Hecke eigenvalues agree on $T_p$ for all rational primes prime to the level. Does it follow that the two forms are the say by some sort of multiplicity theorem?
No, it doesn't follow. Take e.g. the case where F is real quadratic. The space of Hilbert modular forms has an involution coming from the action of the nontrivial automorphism $\sigma \in \operatorname{Gal}(F/\mathbf{Q})$, and this interchanges the Hecke eigenvalues at $\mathfrak{m}$ and $\mathfrak{m}^\sigma$. So the Hecke eigenvalues at rational primes cannot distinguish $f$ from its image $f^\sigma$ under this involution.