No it wouldn't. This has a very different meaning in comparison to "Kateřina says so". In Czech it would be "Kateřina" to říká".
Basically "Kateřina says so" is about the particular content, but "Kateřina talks like that." is about the way of speaking (or about more general content than the former).
You could say "K speaks so" or "K talks so", but not -- as VladimirFu explains -- "K says so".
An accurate translation would be "Kateřina speaks so," where 'so' in its adverb form is intended to mean "thus" or "like that".
I am not sure about the continuous tense. To me the sentence is about the regular way of speaking.