Is it true that there is no absolute morality?

[Nonpareil]

Does it mean something to be a chair?

I only point this sentence out because it indicates a slightly confused position on your part. You are starting to mix the definition of an object with subjective claims of an entity's worth based on how closely it adheres to that definition. This doesn't actually change anything about the rest of my response, but is worth pointing out regardless, because it might clear things up down the road.

After all a chair with a broken leg is a much worse chair than one that is not broken. That seems like a pretty objective way to evaluate the 'value' of a chair.

No. Assuming that an entity, regardless of how many legs are broken or not, or how comfortable it is to sit in, meets you definition of "chair", all claims of one chair being "better" than another are subjective. They all require you the same basis for judgment of worth - in this case, whether or not it stands upright.

What about living things? Does it mean something to be a cat?

Yes. It means "to be a member of one of the species collectively referred to as 'cats'".

Is a healthy Burmese tiger a better example of a cat than a one-eyed, three-legged alley cat with the mange?

Depending on your subjective basis for evaluating which cat is "better", perhaps.

Cats are objects. To evaluate their degree of "catness" seems like a fairly reasonable undertaking. Hmmmm...interesting question, yes?

Not particularly, no.