People generally don't make precisely the same assumptions. People create widely different moral systems with similar assumptions. So no, that's not the case.
No they don't. Two people with different moral systems can still have an intelligible debate about morality. The reason is that our moral systems are not free-standing, complete systems; rather they are reductionist attempts to understand our pretheoretical moral practices.
Without a practice, you don't have a moral theory. You don't have the ability to have meaningful conversations about something without a large degree of shared belief. You don't have a large degree of shared belief unless you have a shared practice that grounds those beliefs and provides the philosophical grammar of those beliefs.
We can have meaningful conversations with each other about moral issues – even if we have disagreements because we share a practice. In other words it is only because we agree about almost everything, that we are able to disagree about details. This is Wittgenstein's lesson: disagreement is only possible against a large background of agreement, because belief and meaning are interdependent concepts.
If you can talk about morality with someone, then that means you share a large background of belief about it. That is not relativism, which assumes that you can talk to someone about morality without sharing any of their moral beliefs. If that was the case, no one would be able to understand each other.
But you can understand people when they talk about morality with you: ergo relativism is false.
As I said: if you bothered reading more recent philosophy, you would see why relativism involves a misunderstanding of the way language works.
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