You're right, Spirodopoulos is seeing something in himself that I don't think he's ever really seen before...Starfleet, with its confused identity between scientific, peacekeeping, and military force, doesn't really quite get that way. I've always had the feeling Spirodopoulos would've been a MACO in another life.
I would imagine both parties see something of themselves in the other. While the conversation between Macet and Spirodopoulos looks tense--I would actually say the tension was NOT necessarily a signal of something wrong, as someone might've assumed if they were just listening in.
But yes, I think for Spirodopoulos, this is a very important moment, because here I think the sheer enormity of what he's done is first starting to register.
(BTW, if you think there's a confused identity going on HERE, just you wait until you read "The Nature of the Beast"!