One reason to believe that this phenomenon is ubiquitous yet not so interesting is the following construction:
Given an equivalence of categories $F \colon \mathscr C \stackrel\sim\to \mathscr D$, define the category of triples $(c,d,f)$ of an object $c \in \mathscr C$, an object $d \in \mathscr D$, and an isomorphism $f \colon F(c) \stackrel\sim\to d$. It has forgetful projections to $\mathscr C$ and $\mathscr D$ that are both equivalences.
For an example of this phenomenon in action, consider the equivalence between Boolean rings and Boolean lattices. It's not really forgetful in the sense that you need to define some new structure going both ways. However, we can define a category of Boolean ring lattices $(A,+,\cdot,0,1,\wedge,\vee)$ expressing both sets of axioms and the relations between them (ok, in this case $\cdot$ and $\wedge$ are literally the same operation). Then both forgetful functors are equivalences.
Likewise, there is an equivalence between algebraic lattices $(L,\wedge,\vee)$ and posets $(P,\leq)$ with finite products and coproducts, and one can consider the object $(L,\wedge,\vee,\leq)$ remembering both. Then the forgetful functors are equivalences.
(In both examples, we only work with $f \colon F(c) \stackrel\sim\to d$ that are the identity on some fixed object $F(c) = d$.)