I think the source of your confusion is improper definition of terms. Let us take a look at this, for example:

If every single ethnic group is a mix of other ethnic groups, how can they be considered only one ethnic group instead of a mixture?

What does this mean? Your question itself is incoherent, because if every single ethnic group is a mix of other ethnic groups, then the category "ethnic group" must necessarily refer to no content at all. It would be a label that does not refer to anything. We would not be able to speak of a "mixture", as there is nothing to mix, since all "ethnic groups" (something that would not exist according to your phrasing) constitute each other. Your definition of "ethnic group" here is self-referential and consists of nothing else other than that self-reference. The fact that the term "ethnic group" exists and carries specific meaning that we all understand already demonstrates that this is not the case. If you go to Europe and look at individuals with Caucasian features, then go to the opposite end of Eurasia and look at Chinese people with Chinese features, it will be very easy for you to tell that you are dealing with two separate groups that have their own separate features. The term "ethnic group" refers to what distinguishes these two groups from each other. If these two groups were indistinguishable, you would not be speaking of a "mixture", but rather would not be speaking of ethnicity at all.

And if everyone is a mixture, how does it make sense to claim someone is mixed now (e.g. "half-white, half black")?

This is also something worth addressing. Let us take White Americans as an example. White Americans are a mixture of various European ethnicities. Would you say that, since they are a mixture of various ethnicities, they are the same or at least indistinguishable from someone who is "half white, half black"? I do not think you would, even though that would be technically correct in absolute terms so long as you are looking at those very specific criteria. This type of observation is useless, though. It does not account for relative difference or context at all and is therefore a lot more nonsensical than the alternative.