I’m not sure that you can strictly say that domestic cats do bite us as a show of affection. It is sometimes a by-product of an affectionate encounter with your cat but I don’t think your cat decides to bite you in order to tell you that he likes you.
There are several reasons why, sometimes, domestic cats do bite during an affectionate interaction with their human companion. One might be that the cat has been overstimulated through petting and starts to play. Play invariably means biting. Therefore this is not a sign of affection except that your cat is playing with you because he/she is comfortable to be with you. But that’s not quite the same thing.
I believe that if a young domestic cat has not been completely socialised or is going through the socialisation process he may tend to bite his human companion when interacting in an affectionate manner. The bite might be quite gentle or might be stronger. Once again it could be play or it might be that the cat is a little nervous with being emotionally and physically close to their human companion and bites defensively or proactively. It may be the behaviour of a slightly stressed cat whose brain is flicking between calmness and enjoying being around a human companion and a more defensive behaviour pattern which involves biting.
I think, generally, that a lot of domestic cats are not 100% socialised. This can lead to biting at any time including when their human guardian is being affectionate towards them. For example, a cat might bite a person’s hand as he or she goes to stroke their cat. The hand is a moving object and a prey item which the cat instinctively attacks. This would be a failure of socialisation.
Dr Bradshaw in his book Cat Sense intimated that one reason for a cat biting his owner might be as an attempt to be dominant towards the owner.
A person on Quora.com suggest that if a young kitten bites you it is about teething rather than anything else. I have not heard of that before but it makes sense.
People often refer to the feline “love bite”. I know what they mean. It is a gentle nibble which is perhaps similar to the human love bite. Indeed it may be a version of the human love bite and one which is truly linked to affection by the cat for their human companion. But are love bites true signs of affection? Is there a conflict between opposing emotions behind it?