In this post, all vector spaces are assumed to be real and finite-dimensional.
First of all, eigenvalues are properties of linear transformations (from a space to itself), while singular values are properties of general linear maps.
Eigenvalues are invariant under matrix similarity ($A \approx P^{-1}AP$ for invertible $P$), which corresponds to describing the same tranformation using a new (not necessarily orthonormal) basis.
Singular values are invariant under orthogonal equivalence ($A \approx P A Q^T$ for orthogonal $P$ and $Q$), which corresponds to choosing different orthonormal bases for the domain and the image.
Real eigenvalues are the real roots of the characteristic polynomial.
Thus, an n by n real matrix can have fewer than n real eigenvalues.
E.g., a 2D rotation matrix generally has no real eigenvalues.
Meanwhile, even with n eigenvalues, the eigenvectors may not be a full basis, as an eigenvalue’s geometric multiplicity (the dimension of its eigenspace) can be less than its algebraic multiplicity.
E.g., the shear matrix \(\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) has eigenvalues 1 and 1, but there is only one eigenvector $[1,0]^T$.
These facts imply that not all real transformations are diagonalizable, or can be described purely as stretching along a basis. In fact, the simplest form under similarity is the Jordan normal form.
As a special case, real symmetric matrices are always diagonalizable (under orthogonal similarity), and all their eigenvalues are real.
In contrast, SVD can diagonalize any linear map because it allows to use different orthonormal bases for the domain and the image.
Also, singular values are always real and non-negative, as singular value of $A$ is the square root of the eigenvalue of the PSD matrix $A^TA$.