Yes, yes, yes - as it occurred to me in a sleepless night, you are absolutely right, sorry Gianluca! The correct result of
NIntegrate[Sqrt[1 + Exp[-2 x]], Element[{x, y}, Circle[{0, 0}, 2]]]
should definitely be 0. And one can generate more questionable results (at least with my version 13.3), like:
Integrate[n, x \[Element] Point[{2}]]
(* Out: n *)
Integrate[1, x \[Element] Circle[]]
(* Out: 2 \[Pi] *)
Integrate[1, {x, y} \[Element] Circle[]]
(* Out: 2 \[Pi] *)
Integrate[1, x \[Element] Disk[]]
(* Out: \[Pi] *)