Can we think of an adjunction as a homotopy equivalence of categories?

There is a way in which we can think about a natural transformation $\eta: F \rightarrow G$ as a homotopy between functors $F,G:\mathcal{C}\rightarrow \mathcal{D}$. Now, an adjunction $F \dashv G$ gives rise to "homotopies" $\epsilon:FG \rightarrow 1_{\mathcal{C}}$ and $\eta:1_{\mathcal{D}} \rightarrow GF$, namely the counit and unit. If again we borrow the language of topology, one could say that the existence of an adjunction implies that the two categories $\mathcal{C}$ and $\mathcal{D}$ have the same "homotopy type".

Is this a useful picture to have in mind when dealing with adjunctions? Is it well-known?

Thanks for your time.

Yes. An adjunction between two categories gives rise to a homotopy equivalence between their nerves.

Also observe that if $I = \{ 0 \to 1 \}$ denotes the interval category, then a natural transformation between two functors is precisely a functor $C \times I \to D$. So natural transformations are like homotopies between functions in this sense, except that they are "directed" in a way that homotopies aren't. This idea gives rise to directed homotopy theory.