How can Chomsky hierarchy be applied to languages with alternated letters?
Every regular grammar produces a regular language.
But:
Not every grammar for a regular language is regular.
In other words, a regular language can be described by any of an infinitude of grammars, some of which are regular and some not.
In this case, you can easily create either a left- or a right-regular grammar for your language. For example:
\begin{array}{l}
\mathrm{S} \rightarrow \mathrm{X} \mid \mathrm{Y} \\
\mathrm{X} \rightarrow  \mathrm{X}  \mathrm{a}\mathrm{b} \mid \mathrm{ab} \\
\mathrm{Y} \rightarrow \mathrm{Y}   \mathrm{b}  \mathrm{a} \mid \mathrm{ba}
\end{array}
Of course, that transformation only preserves the language parsed, not the induced parse trees.