1. e4 e5 2. Ng1f3 Nb8c6 3. Nb1c3 Ng8f6
(four Knight opening)
4. Nf3xe5 {!?}
I see two options
4b. Nc6xe5 {leads to complications after 5w. d5, as seen in linked game }
4b. Nf6xe4 {How manyh moves can black play 100% symmetrical ?}
Game 766955
Well I would play Nxe5 and take the material advantage. After d4 play Ng6 and avoid the complications. Say this line:
From your given position
4. ... Nxe5
5. d4 Ng6
6. e5 Ng8
7. Bc4 d6
At this point just trade for the endgame. If your opponent wishes to play this handicapping opening then let him. Just beat him and maybe he'll realize the errors of his ways.