Decide if a given sequence is regular or context-free
As you already noted, a single example or any finite number of words in $L$ is not going to be the right notion. So why not look beyond? A somewhat famous paper did so in 1967 and revealed what happens when you try to learn in the limit.

An alternative interpretation of your question leads to the field of descriptive complexity theory where one looks for a minimal representation $\langle M,w\rangle$ of a string in terms of a TM $M$ and an input word $w$ to it. Asking whether a single word had been (better?) generated by a CFG or a DFA is indeed ill-posed: either one would be a special case of such a pair $\langle M,w\rangle$.

Finally, given $w\in\Sigma^*$, asking for the smallest DFA $A$ and CFG $G$ with $w\in L(A)$ and $w\in L(G)$ is going to disappoint. The simplest DFA (with just one state) and the simplest CFG (with one non-terminal) will do because the universal language $\Sigma^*$ is blatantly regular. To make this intersting you'd need some negative instances, which brings you back to reading said famous article. I recommend it.