I came across the following statement in a paper,
The hyperbolic Laplacian is a real smooth operator, and the system of nodal lines determines $f$ up to a constant multiple.
Here, $f$ is an eigenfunction of $\Delta$ on the hyperbolic plane. It is not clear to me why the second part of the above statement is true, i.e. why two eigenfunctions of $\Delta$ must be scalar multiples if they have the same nodal set.
I believe it more for eigenfunctions which are orthogonal, but there are some facts about Laplace eigenfunctions that I can't verify, probably due to my lack of PDE knowledge. If we knew somehow that it suffices to consider real-valued eigenfunctions, and if we knew somehow that an eigenfunction must change sign when moving between adjacent nodal domains, then it must be true that orthogonal eigenfunctions cannot have the same nodal set. For the first point, perhaps breaking an eigenfunction up into its real and imaginary parts will help, but these real and imaginary parts don't necessarily have the same nodal set. For the second point, I think this is equivalent to $f=\nabla f=0$ having only finitely many solutions, which sounds plausible, but I don't know how to prove it. References to books that cover properties of eigenfunctions and nodal sets well (and from scratch) would also be great.
When people talk about nodal lines, it's understood that eigenfunctions are real-valued. (Otherwise, the statement would be false already for an interval, where distinct eigenfunctions $\exp(i nx)$ share the same, empty, nodal set.)
Pick any nodal domain $\Omega$. The restriction of eigenfunction to $\Omega$ is a constant-sign eigenfunction for $\Omega$, i.e., the one for the principal eigenvalue. Such an eigenfunction is unique up to a constant multiple (the lowest Dirichlet eigenvalue is simple). Since eigenfunctions are real-analytic, uniqueness holds everywhere.