Why does it matter that they are capable of suffering? This is a serious question. When it comes to morality invariably when you dig down deep enough, all of it is based on human made assumptions. That doesn't invalidate discussions about morality, we just have to be aware that it is built on a subjective foundation.

And for the record, I personally suggest a vegan lifestyle, because I believe exploiting animals is morally wrong.

Because what stands in the center of it is empathy, which is not a human only thing

Cruelty, antipathy and apathy are also not uniquely human. Yes, you will see some animals behaving empathically in some situations, but people that walk over to pet bears and get mauled don't really earn much sympathy from people who know wild animals. Any given animal species will not share our norms. As an extreme example, the guinea worm should be extinguished from this earth. I believe no one can empathize with that species, but in any case I will support those people who seek to make it extinct. Between that extreme and equal membership in society are many models of co-existence and we will naturally disagree amongst ourselves of what norms we should have regarding that co-existence. There are no universal norms for our co-existence.

Hey, good for you for asking this question despite being on the animal rights side. I wish I could say something useful as an answer. I could snipe at Jeremy Bentham for kicking it off in 1780 with shoddy reasoning: not, Can they reason? nor, Can they talk? but, Can they suffer? he says, but this is really a side effect of an argument against slavery and contains no discussion of why reducing suffering is desirable, and he was also responsible for utilitarianism, "the greatest happiness of the greatest number", a fundamentally awkward fundamental axiom - and happiness isn't even identical with the things we tend to think are right - and then you have the problem of animals causing suffering to other animals routinely. But I can only agree that it's a good question.

What makes it a good question? It's obvious that we can't conjure universally compelling moral axioms out of nowhere; all of our moral reasoning ultimately rests on fundamental values that one either cares about or doesn't. If our sceptic has the capacity for empathy with animals, we can try to convince them to elevate their empathetic response to the status of a moral value. But if they simply don't care, and insist that they will only begin to care if logically compelled to do so, we're in the same position as if we were trying to reason a sociopath into valuing other people's wellbeing as an intrinsic good.

I think, first of all, you're not wrong, but also, you're mostly wrong. People with completely separate value systems exist, but only theoretically, or historically. I know there's a lot of hand-wringing at the moment about society being fractured and divided, and it may be Pollyannaish of me to say this, but I think we're mostly all on the same side with approximately the same bunch of deep values, by the power of culture (and we more or less all come from the same culture, "modern"). We can have productive discussions, we can work it out. The Myth of the Framework is a relevant book.

To "prove" is to derive from a set of consistent axioms. There is no set of universally agreed axioms for badness and in fact the vast majority of people do not even have a consistent set of moral beliefs. Simply changing how you phrase a question can get very different answers (e.g. "conscription" vs "murder slavery" vs "making people defend the country that defends them").

Yes. This doesn't mean that morality is subjective and divorced from reality, mind you. It just means that trying to hone an objective theory of morality will make you unpopular.

Depends what definition of "morality" you're using. Most are subjective and the remainder tend to be either descriptive, local (making use of the speaker, and their properties such as their culture, in their definition/meaning) or religious.