The analogy is of course a correct/useful analogy, but I think any model structure for which the literal statement is correct must have $Set_* \simeq *$, so it would be a bit too coarse to do anything reasonable with categories *as categories*. As Alexander Campbell points out in the comments below, in the Thomason model stucture (which uses categories to model homotopy types) we do have $Set_* \simeq *$, but it destroys (what I guessed is) the point of you question.

Indeed, your assumption guarantees the existence of a homotopy pullback square: 

$$\require{AMScd} 
\begin{CD}Set_* @>>> * \\
@VVV @VVV \\
Set @>>> Set \end{CD}$$

But $Set \to Set$ is also a weak equivalence (it is an isomorphism !), so 
$$\require{AMScd} 
\begin{CD}* @>>> * \\
@VVV @VVV \\
Set @>>> Set \end{CD}$$
is also a homotopy pullback, which implies $Set_*\simeq *$. 

Pseudo-limits (the kind where the diagrams commute up to isomorphism rather than just $2$-cells) can be expressed as homotopy limits, though. But for this kind of (op)lax limit, it seems like one need some notion of $2$-model category to make it work. I don't know more about this though.