Is SVD non-linear while PCA (by eigendecompostion) is linear?
To the best of my knowledge no.

SVD and PCA are both linear dimensionality reduction algorithms. Some nonlinear dimensionality reduction algorithms are e.g. LLE, Kernel-PCA, Isomap, etc.

About t-SNE I would like to add a point. It reduces the dimensionality (and does it pretty well!) but it is only for visualization and can not be used in learning process! So be careful putting all these next to each other. In other words, they are all dimensionality reduction algorithms however, PCA and SVD can be used for feature extraction but t-SNE can not. All can be used for visualization purposes (in EDA).

I certainly recommend reading this answer. Probably the fact that "the square roots of the eigenvalues of $XX^⊤$ are the singular values of $X$" confused your friend that it's a nonlinear method.

Hope it helps. Good Luck!