When you turn this picture over, it says 12/22 (most likely of 1967). That is the last known instance we see Adler without his scars.
Then the first known instance we do see him with his scars is in this cutscene with Stitch.
It's implied that the interrogation took place after Rebirth Island, but that's simply not possible for his scars to look as healed as they are—if it happens right after, at least. And we're never really given a proper date to when it does.
What we do know is that Stitch was contacted by Perseus in 1972 in Petropavlovsk Gulag, which gives a 4 year window between that and the Rebirth Island Raid; for Stitch getting interrogated, escaping, punished by Kravchenko and sent to the Gulag. My bet? '69, since it gives Adler a year to heal, and a plausible time period of Stitch being held in custody.
And bonus, we do get these undated photos of Adler in the Fracture Jaw briefing cutscene.
[Credit to @adlerboi]
In fact, if I could keep going, I'd go as far as theorising that he got his scars in the fall of MACV-SOG. Yeah, the mission you play in BO1.
These photos are of a very scarred and very cursed looking Adler way back in Vietnam itself. Meaning that's where he has to have gotten them. That's it; ladies and gents and non binaries, we got em, lock it in.
C'mon, Adler not being called in to defend as literally MACV falls? Especially when he's in such close proximity? Mason even recollects it:
We were after a different spy (the defector) in KheSanh. That's right. Our team was right next to Adler's when we touched down in Hue City.
Maybe he was also being literal; maybe he was saying that Adler's unit was literally next to theirs right there, defending a different part of the compound while they went in to secure the defector.
But even if he isn't, I find it hard to believe that Adler wouldn't be there to defend from the compound being overrun. We don't see him—cause he wasn't created yet—but it doesn't mean he couldn't have been in a different part of the same conflict.
Also, the Rebirth Island Raid happens right after—which he was part of, and the following Nova 6 crisis. And then his work with Stitch, chasing Perseus, etc etc; so this was likely his last operation in Vietnam, which makes also makes it the last opportunity for him to get scarred.
Why I'm fixated on just this idea is cause it'd track with one of the stories Adler tells us in BO6:
Got hit by shrapnel in Hue City—this was back in '68. Hurt like hell.
The details check out, and there really aren't many more opportunities for the story to happen in that timeframe. Plus, it does feel like Adler to drop the actual story like it's a fib.
There was also the possibility of it being from Fracture Jaw, which could also make sense; Perseus being the indirect reason he got his scars is pretty neat and gives Adler a reason to call him his lifelong nemesis. But me personally, I doubt Adler would give Bell his memories of that very vulnerable event for the sake of a memory exercise. And I don't really think Fracture jaw was that massive of a fuckup; the helicopter crashed—which is about as common for COD characters as finding a three leaf clover—and they had to defend from VC for like 5 seconds till Napalm came in. Doesn't really feel that plausible, but it's a possibility, I suppose.
I still really like my own hypothesis, inaccurate as it may be. Adler's called in for reinforcements at Hue City, he gets overrun, tanks were rolling in and a strike next to him got his face in the ongoing mayhem. Air support makes it in time to rescue the wounded, saving him, but the scars leave their mark permanently. Literally, and figuratively.
Conclusion: Adler got his scars in early '68, in Vietnam.
