They aren't false friends. At best they are partial friends. Familiar in Spanish does also mean what it means in English. But we have changed a single letter to make our adjective that is like this one, although it has altered its usage somewhat. Our word is familial. A lot of the cognates labeled false friends are actually partial friends. Even embarazo is in the DLE with the meaning of embarrassed, although I haven't seen it used. And others are mostly friends, but do have one small difference. That's why you will see me talking about a perfect cognate, those words that we understand perfectly how to use in Spanish, because we do in English.
No. Remember that in English nouns can act like adjectives and modify each other. But in Spanish you either need an actual related adjective or some sort of prepositional phrase generally with de. Familiar is the adjectival form of family in Spanish.
If you look up the word familiar, it can be translated either way. so context would be needed to honestly know which is which. Without context, Duo should not penalize users for not knowing the context... this is one of the few examples where that is the case - normally you have to "know" the context on Duo.
I agree that both familiar discussion and family discussion should be accepted and that as a Spanish phrase family discussion is probably a little more probable. But to suggest quarrel as a synonym says a lot more about your family's discussions or at least your feeling about them then any linguistic point. I am not even sure if I would consider all quarrels discussions, but certainly all discussions aren't quarrels.
Just because everybody here wants it to be the opposite of what it is doesn't make it so.
The spanish word for argument is discusión.
Discusión familiar does not mean a familiar discussion.
Discusión familiar =
Although it is possible it could be as mild as a rousing debate.
I notice that DL gives discusión as meaning either discussion or argument. However, I am guessing that discussion is generally more closely matching, given that they sound similar. A discussion is a more civilised, harmonious exchange of different ideas between two or more people. An argument is more of a conflict of words between two or more people.
I have read many of your very excellent comments. However this is not one that I can describe that way.
You have guessed wrong where I have never observed you to guess at all before.
Necesitamos tener una discusión familiar, jovencita.
We need to have a family discussion, young lady.
Please notice that the tone of both the above and below sentences is that of a parent beginning a disciplinary session.
Es hora de que tengamos una discusión familiar.
It's about time we had a family discussion.
tener una discusión =
to have an argument
After the Norman invasion in 1066 a variety of French was spoken in schools Universities, courts and among the upper classes almost exclusively for about 200 years in England. When English was again the major language of the land, it was Middle English which had significant input from Latin. Old English is very much a Germanic language.
Familiar is both a Spanish and an English word. As a Spanish noun it means relative. As a Spanish adjective it has various meanings including both relating to family as well as a true cognate to our word familiar. Here is the definition and examples from Spanishdict.com
Many words have diverse meanings. Familiar is ethomoligically related to the word family. You are familiar with things that your family does. But we generally use familial as related to family, but the change has been in English. Familiar in Spanish does mean both. Here is the Spanish definition of familiar, which may be less confusing.