6:05 2Z6 is not a good example because it does have an identity, 4 is the identity because 4*4=16=4(mod 6) and 2*4=4*2=8=2(mod 6) A better example would be 2Z8.
I got 0:17 seconds into this video before I started screaming. Thankfully it looks like there's a separate video explaining zero divisors. Bless, we live to math another day.