Me thinks the older Admiral was a bishi in his prime until the reality of war hit him and he lost his eye in battle. He's probably trying to impart some wisdom on the younger admiral before it's too late.
I see the younger one's actual rank is Lt. commander. So do ship girls just call every naval officer "Admiral"?
Japanese has different words for the rank of Commander and and the position of Commander (commanding a squadron/division) and Admiral (commanding a fleet).
The words that the shipgirls use 司令官 and 提督 refer specifically to the position. In the IJN, (full) Commander (OF-4) was 中佐 (chuusa) and (full) Admiral (OF-9) was 大将 (taishou).
The JDSF uses 2等海佐 and 海上幕僚長たる海将 respectively.
This is why the destroyers, which operated in smaller units such as the 6thDesDiv, use Commander, as that's what the real commanding officer would be called; the same with the bigger ships, as they tended to only operate in larger fleets, which would be commanded by a Admiral (with rank of whatever).
In the real world anyway. In the Kancolle world, you have 1 Harem leader commanding over 100+ ships so it's easy to see the confusion some people would have of "why are all the characters calling him by different names?"
1 minute you hae some destroyer Loli's being all "Commander commander praise me!" while the next minute, you get a battleship going "Admiral! Take me now!"
Urk.Tenryuuuu...ahahah. So you got scolded by your senpai♡Well, it's certainly a nice thing to have self-confidence, but...You're not gonna freakin' win with your power alone, you dumbass!There, there'ya totally like a kid, there.I'm back to base!
... the heck? 'ya hat's on deeper than usual, innit?This kinda Admiral is nice too...Lt. Commander, huh? Aren't you just gettin' ahead in life?The line between confidence and conceit is only paper thin, you know?Do take good care of your shipgirls, okay?
Congratulations on your promotion.That guy is so terrifying when he's maaaaad...Siiir.'ya better make damn sure that 'ya don't lose something important when yer young, like I did, 'ya know?
Pretty-boy Lt. Commander.Nah, it's more of a feeling of 'I finally got here'.