Why is this authenticated Diffie–Hellman key exchange insecure?
My guess is that when it is written approximately

Can we really have a non-replayable 2-msg protocol?
□ Remember (the question's protocol) insecurity

the author is thinking about an attacker capturing the first message of the exchange $(A, g^x, \operatorname{SIG}_a((g^x, B)))$ and using a replay of that to impersonate Alice.
There are two variants:

Declaring victory just because Bob thinks (and the logs of its machine show) that Alice connected again, when she did not.
Further, if Alice leaked the $x$ she used in the past, that allows the attacker to get the new shared secret by doing just as Alice normally does, and fully impersonate Alice.