"Each person can count."
Translation:Każdy człowiek umie liczyć.
No. "Każdy" is a pronoun that behaves like an adjective - it changes with cases and also in gender. It is used for singular. "Wszyscy" could be considered it's plural equivalent. It also changes with cases and gender.
"Każdy" generally assumes that it is "każdy człowiek", "wszyscy" assumes that it is "wszyscy ludzie". But you can also have "każda kaczka", "każde dziecko", "wszystkie kobiety", etc.
So only the second of your sentences is right, the first should use 3rd person singular form "umie".
"umie liczyć" = he/she/it can count = has the ability to count.
"mogę liczyć" = I can count = I am allowed to count, something like that... it's not that probable that someone will say that. Maybe if there are a few tasks to distribute between people and someone says "I can count".