Is intersection of regular language and context free language is "always" context free language
The claim is that the intersection of a regular language and a context-free language is context-free. You've intersected a regular language ($\{ab\}$) and a context-free language ($\{a^nb^n\mid n\geq 0\}$) and the result was a context-free language ($\{a b\}$). Sure, that language is also regular but every regular language is also context-free. The statement isn't that the intersection of a regular language and a context-free language is a non-regular context-free language.
Analogously, the child of two mammals is a mammal. You've taken two mammals and you're saying, &quot;Why is their child a cat? It's supposed to be a mammal.&quot;