Something that has confused me for long is the condition for injectivity for a homomorphism. The condition is that the kernel should just be the identity element.

I used to think that maybe this condition for injectivity applies to all mappings, and wondered why I hadn’t come across this earlier.

No. This condition applies only to homomorphisms $f:(G,*)\to (H,.)$ due to their special property: $f(a*b)=f(a).f(b)$. This condition would no longer be valid if the mapping was so defined: $f(a*b)=c.f(a).f(b)$, where $c\neq e_H$.