I would say yes, to a limited degree, for a limited time, sometimes.
So I have a number of outdoor cats, and there was one time I went outside and one of my cats had died. Idek from what. His brother was just right beside him, sniffing him, trying to get him to move. I was like ooooh, this isn't good. I quickly grabbed the dead cat & moved out of sight, buried him.
But that damn cat wouldn't let it go. For a good couple days, he just had this thousand yard stare & a sad moaning meow. He discovered death, he seemed to realize death applies to everyone. Himself included. It reminded me of my own epiphany when I was 7 or 8, I almost wanted to chuckle a bit, it's like yeah bro welcome to the club. Death is our shared fate. Just try not to fixate on it.
So my cat's limit was about 2 days. After that, he seemed to move on & he was back to being happy again. Some extra seafood, catnip, and pets helped coax him out of the slump.
Of COURSE they do. Hell they feel existential dread towards things that aren't even there. Its just that their body language is so different from ours, that we don't usually recognize it.
Hell just staring into an unfamiliar cat's eyes can give it a case of the screaming jibblies.