The closest that I know of that's known and possible is with a fungus. Possibly a parasite, but probably fungal.

Some parasites, especially those that infect insects, can cause the host to go crazy, for lack of a better word, and "infect" others through bodily fluid transfer. Fungal is similar; IMO, fungal would be my guess since to me, given how large fungal networks and organisms can be, it would be the most likely candidate to adapt to the size and scale required to control a human.

However, it is unlikely. There's a pretty slim margin between being infectious enough to be viable and so infectious the host dies before there can be any useful progression of the disease. It's just a very fine line.

Depending on what version of zombies you're thinking of, it may be more, or less zombie-like. In the case of the walking dead? No. Not really possible. Maybe in a million years, caused by nano scale machines, where the machines more or less use your corpse as meat armor... But that's a very long way from becoming a reality. The "real zombies would be more like possessed living people, still vulnerable to the same dangers as other living humans. If you shoot them, they will die. They wouldn't be super strong, maybe mildly more strong than they were when they were living, simply based on the fact that we tend to hold ourselves back a bit when it comes to our strength because we want to avoid damaging ourselves too much during the effort.... "Zombies" wouldn't have those concerns so they may be stronger, but not so significantly that it would matter all that much.

Skin appearance may be affected due to the infection and may cause the flesh to appear sticky.

Since most higher brain functions would be suppressed, tactics and planning would not be possible, or at least, extremely limited, and most advanced skills would also be unavailable (working tools or machinery). There would also be very little in terms of language skills, if any.

Since the infected would have the primary goal of spreading the infection, they likely wouldn't eat or sustain themselves in any meaningful way, leading to death in a matter of days, maybe a week or so, at most. Zombies would probably smell of human excrement, since the infected wouldn't be concerned about where they relieve themselves and likely just piss or shit right in their pants as the need arises.

TL:DR: they would be far shorter lived and far less dangerous than seen on TV but it's possible that a parasite or fungus could invoke such tenancies