Does "unfounded" refer rather to the lack of arguments for a statement or their irrelevantness?
Unfounded has two (similar) meanings:
1. (of ideas, allegations, etc) baseless; groundless
2. not yet founded or established
As such, it can be used to describe statements that are either not based on facts and evidence (1), or statements relating to an idea that is not well established (2).
For definition (1), this means statements such as:
"The sky is made of soup" could be described as unfounded - as there is no evidence of this, and it's not based on factual evidence.
This is fairly straight forward - any statement that either hasn't got enough evidence to be disproven, or any statement whose evidence can be directly refuted - may be described as "unfounded".
For definition (2), this relates more to your question about names.
"PoliticianA is a thief" when referring to them avoiding tax (and not actual criminal theft), may be considered "unfounded" if there is no well established norm for considering tax evasion similar to theft.
Perhaps more clearly, "your dog is evil", may be considered an unfounded belief - even if there is evidence for it hurting people - as there is no well established basis that dogs have a moral compass (and therefore cannot be good or evil).
On a site note, when it comes to the difference between having unfounded beliefs and being a liar. Most people will consider a liar somebody who actively tries to decieve. As such, if the person was not relying on unfounded statements on purpose - it would generally be considered inappropriate to call them a liar.
(The belief that the person is a liar, may itself be considered unfounded - if there is no evidence they made those statements knowingly, or if they are otherwise of good character and so the idea of them being a liar is not well established)
I will try to answer the question from the perspective of how to use the word unfounded in a sentence. What can be said to be unfounded?
Following up on a productive distinction made in this comment:
Ok, so that is not what I meant. What if there is evidence for something but the term is inappropriate? Like calling an entepreuner a thief if they registered their enterprise in a low-tax country (it's legal). Or calling someone a rapist for touching someone, even though it has been caught on camera.
The names themselves, the names per se, are not able to be unfounded, nor is the act of name-calling itself able to be unfounded. Rather the underlying claim or assertion will either have a basis in fact or be unfounded.
So it would be awkward to say Calling me a liar is unfounded. Calling me a liar is unwarranted or unjustified. It would be better to say Your claim that I'm a liar is unfounded. or That remark was unfounded. The assertion itself is what has, or does not have, a basis in fact. The act of making the assertion is not itself able to be factual.
But in conversation you could retort:
"Liar" is unfounded!
because the intended (elliptical) meaning is "the allegation that I'm a liar...".
P.S. If the question is Can an exaggeration be "unfounded"? --well, yes, it can be, or at least the kernel assertion can be, because an exaggeration contains an assertion.
P.P.S. unfounded does not mean "having an insufficient basis in fact".