Regular languages under change of encoding
The answer is no.
I'll give an example of a language $L$ which is regular in binary but not in unary:

Consider $L=\{10^k|k\in \mathbb{N}\}$. The corresponding language in unary is $L'=\{1^{2^k}|k\in \mathbb{N}\}$.

It's easy to see that $L$ is regular while $L'$ is not even context free.

L'' also isn't regular either, by the link @Sylvain posted in his comment.