NP and finite automata
What you have given is a definition. You can define anything you want.  The class of languages defined as


  A language $L$ is in UC (user71565's class1) if there are a polynomial&nbsp;$q$ and a DFA&nbsp;$A$ such that $x\in L$ if and only if there is some string&nbsp;$y$ with $|y|\leq q(|x|)$ and $A$ accepts&nbsp;$xy$


is a perfectly legitimate class of languages.  For example, it includes all regular langauges (take $y=\varepsilon$ for every&nbsp;$x$).

However, because this is not the same definition as the definition of NP that you quoted in your question, you shouldn't expect that UC$\,=\,$NP.  In fact, we can quickly see that UC$\,\neq\,$NP, using the pumping lemma and a language such as $\{0^n1^n\mid n\geq 0\}$, which is in NP but cannot be recognized by any DFA.

Interestingly, though, the following definition is equivalent to the normal definition of NP:


  A language $L$ is in NP2 if there are a polynomial&nbsp;$q$ and a deterministic logspace Turing machine&nbsp;$M$ such that $x\in L$ if and only if there is some string&nbsp;$y$ with $|y|\leq q(|x|)$ and $M(x,y)=1$.


That is, we can weaken from a deterministic polynomial-time verifier to a deterministic logspace one, without changing the class of languages defined. This is a consequence of Fagin's theorem, which says that the verifier can actually be a formula of first-order logic.  The expressive power of first-order formulas is strictly weaker than deterministic logspace Turing machines.



1 There's something of a tradition of naming language classes in this way. For example, the class NC of languages decidable in polylog time with polynomially many processors stands for "Nick's class", after Nick Pippenger.