after St. Louis police unions instructed officers to publicly display The Punisher's insignia (the mark of a lawless, fascist murderer) the comic book community was quick to point out the stupidity, and the frankly horrifying message sent by supposed peace officers endorsing a maniac. So it's a good thing The Punisher personally confirms he hates cops who see him as a friend… making every cop who "wears his mark" or calls themselves a "fan" look like shameful fool in the process.
-- Andrew Dyce, The Punisher Confirms: He HATES Cops Who Support Him
I mean, to me the meaning of that juxtaposition is pretty clear.
The Gadsden flag highlights individual primacy, but the thin blue line sticker makes it clear that it's his individual primacy that he's concerned with. For an anti-authoritarian evoking that symbolism, the 'me' refers to the general autonomy of humanity or at least Americans, but in this case it probably literally refers to that specific individual's autonomy or to the autonomy of the United States as a country in a nationalistic sense.
He's basically just representing his subculture and thumping his chest about how nobody better tell him what to do or get in his way, while also showing that he's affiliated with a big gang. Whether he's aware of the racist speech the symbol is referencing or the symbol's deeper meaning is kind of up in the air, but it still probably wouldn't produce much conflict with his sense of nationalist autonomy in an authoritarian context regardless.
Honestly, it's that context that I think makes the association with Punisher inevitable. Whether the character supports the current gang in charge or not, he clearly believes in an authoritarian model of crime and punishment; that's the lens he views the world through and the impetus for his actions. If it's satire, it certainly doesn't read that way. Though, to be fair, the show is probably a lot more egregious in that regard than the comic (while also likely being more widely consumed).
Frank Castle supports authoritarian measures so much that he goes beyond what the legal system allows for. He literally names himself after an action designed to reinforce authoritarian hierarchy. Sounds pretty on the nose to me.