Is there any mathematical proof of fractal self-similarity?

You can only prove selfsimilarity for fractal sets $K$ that are defined mathematically, e.g., the Cantor set, the Sierpinsky carpet, Koch's curve. In these cases $K$ is not only similar to a part of itself, but $K$ is in fact the union of (more or less disjoint) similar copies of itself: $$K=\bigcup_{i=1}^m f_i(K)\ ,$$ whereby the $f_i$ are similarities. In the above examples this is evident by inspection. All these examples go under the heading Iterated Function System. See the literature quoted in the linked Wikipedia entry, notably Barnsley and Falconer.

If you require only that $K$ is similar to a part of itself (as in your quote from Wikipedia) then there are examples which you might not consider "fractal", e.g., the logarithmic spiral $$r=e^\phi\qquad(-\infty<\phi\leq0)\ .$$

There is no way to demonstrate that property mathematically because the property does not hold. Not all fractals are self-similar and not all self-similar objects are fractals. The rigorous definition of a fractal is an object having a Hausdorff dimension that is less than its topological dimension.

This post asks a similar question and the selected answer gives more detail: Why must fractals be self-referential?