maybe because somebody is asking you if you know a horse which you once saw.
Can you say "pamiętacie tego czarnego konia?" Would this make grammatical sense?
why do we use "tego" in the sentence? is "pamietacie" a verb that requires the use of the genitive case?
It is Accusative here. For masculine singular, there are two possible Accusative forms of "this": "tego" (for animate nouns) and "ten" (for inanimate nouns).
There is a problem because quite a lot of nouns, mostly food and beverages, are 'animate' for no reason, but luckily here there's no such problem.
Well, there are a few others that can't use both Present Simple and Present Continuous. PC "remembering" would be "wspominać tego konia" or similar. "pamiętać" is just to simply have this horse somewhere in your memory where you can possibly access it.