Likely Faye is super impaired during that brief flash of awareness. She's probably mad at the world (Dora fired her, Marten's off having romantic adventures) and not in any condition to look at those feelings rationally. She might recognize that she's in a hospital because of Marten, she might not.
I do think it's meant to be comedy. But it's dark comedy. Unexpected punch, but expected puncher. Bad timing. It's an incongruous moment in a dark strip. There isn't supposed to be deep meaning about it, and I think sober Faye would be pretty mortified, if she weren't likely to wake up to a steaming hot plate of "Why me?"
She's definitely not in her right mind, but then she hasn't been since her dad died.
Someone (BenRG, I think) said Marten's growth has been coming along since Padma. But Faye's meltdown has been coming along just as long. Her attempt to sass Marten about the hiking "date" went somewhere she didn't want, and while it's played for laughs, her fear that Marten might chase Padma to LA is a bit of an overreaction given the situation.
It's looking more an more like Faye didn't break up with Angus because long distance is hard. She did it out fear of abandonment. All of the choices she's made (major ones) come down to giving into that fear or wrestling with it. Her over reaction to getting fired is probably another aspect (Dora abandoned her--not really, but to a person with a pathological issue around abandonment, it would look that way).
We'll have to see. Maybe this strip signifies Faye looking death in the face. I'm still unconvinced she's an alcoholic. A scare like this would certainly jerk someone who isn't out of the bottle. (If she is, it wouldn't. The only medical scare I've seen work, to any degree was "well, your liver has stopped working." He was "lucky" in that he actually had an autoimmune disease, and his drinking was just helping it attack his liver. He regained partial function after he stopped. At the same time, I keep waiting for the news that he fell back into the bottle and drank himself to death.)