The question (with the specific "attack") has been answered in detail in Sec 1.5.6, Attack/ Figure 1.6 .
Essentially, it depends on the specific definition of "mutual authentication", or more generally, authentication goals.
In the cited work, the definition of "mutual authentication" (Def. 14 below) is reached, since both parties can verify that the messages originate from the respective party.
For example, A knows that the message $g^y, E_K(S_B(g^y, g^x))$ come from B by verifying $S_B$, and thus that B has knowledge of the content.
However, the definition of "strong entity authentication" (see Def. 13 below) is not met, since
"A would be wrong to conclude, after a successful run,
that B wishes to communicate with her." Sec 1.5.6
"Definition 13. Strong entity authentication of A to B is provided if B has a fresh
assurance that A has knowledge of B as her peer entity" Def 13., Strong Entity Authentication
"Definition 14. Mutual authentication occurs if both entities are authenticated to each
other in the same protocol. Unilateral authentication (sometimes called one-way au-
thentication) occurs if only one entity is authenticated to the other." Def 14., Mutual Authentication