If a machine recognizes language L, can it also recognize L*?
One problem I see with this is that a machine can only accept one language, so it might be problematic.


You are answering your own question: by definition¹, any given automaton accepts $L$ and $L^*$ if and only if $L = L^*$. This can happen -- examples include $L = \{ \varepsilon \}$ and $L = \Sigma^*$, for instance, but not as a rule.

It is not clear from your question if you understand the acceptance criterion of finite automata. The automaton has to be in a final state after reading the whole input, not in the beginning or at any time.




The language accepted by automaton $A$ is uniquely defined to be the set of words $w$ for which $A(w)$ is (an) accepting (computation).