What would be different if you wanted to say "who knows your father?" rather than "whom does your know?" ?
Kto zna twojego tatę/tata (you need accusative after zna, and tata declension is like mama(acc = mamę, tatę), tato has weird declension (acc=tata))
Kogo is acc of kto. A way to think of it is 'your father knows whom' it looks odd but in effect thats whats being asked.
It's actually the main answer.
And actually, it shouldn't be. We've had the who vs. whom discussion dozens of times, and 'who' has won. I don't like it myself, you don't have to persuade me. But those natives that think that 'whom' is dated have won. Let's just leave it here.
I'm not a native English speaker but I think that it's more used who knows your father? Can you explain me the differences, please?
The sentence is written the other way round, this one may be less common but we wanted to show another grammatical construction:
"Kogo zna twój tata?" (Whom does your father know?) has "your father" as the subject. The answer is: "My father knows David Beckham!"
"Kto zna twojego tatę?" (Who knows your dad?) has "who" as the subject. The answer is: "Mr. Johnson knows my dad."