Wikipedia explains that pleonasm is the use of more words or parts of words than are necessary or sufficient for clear expression: examples are "black darkness", "burning fire".
I believe this applies very well here, with the caveat that I think you might be misinterpreting silent and using a definition that is not the one intended by the speaker: the word silent, when used to qualify people, doesn't have to literally refer to someone who never says a single word throughout the course of their life (that would be "mute", if anything!), but merely that they are taciturn or not loquacious – which is not a binary description.
"He is silent" could be employing either meaning, depending on whether the context involves him being silent now, or him being a generally silent person; when I see the statement "He is extremely silent", though, I automatically assume it's a description of the person's general quality of being very taciturn, and as such, not really a pleonasm at all.
Something that was pointed out in comments is that "pleonasm" is often employed more narrowly to refer to the use of multiple words that have related meanings, and even Wikipedia calls pleonasm a type of tautology, and provides examples of that type; however, the basic working definition it presents is "the use of more words or parts of words than are necessary or sufficient for clear expression". If we stopped at that, "extremely silent" would qualify (minus the above caveat).
Since just picking and choosing parts of a Wikipedia definition isn't satisfactory, I will also look elsewhere.
Merriam-Webster defines it as
the use of more words than those necessary to denote mere sense (as in the man he said)
and ODE as
the use of more words than are necessary to convey meaning (e.g. see with one's eyes), either as a fault of style or for emphasis.
Admittedly, both examples involve repetition of related meanings ("he" and "the man"; "see" and "eyes"), but the definitions themselves don't. I think the examples can be explained away as the most obvious type of pleonasm that comes to mind.
I find it interesting that one of the dictionaries provided by Dictionary.com (the "British Dictionary") actually gives two separate definitions, the first being similar to the ones we've seen, but the second being a terser
a word or phrase that is superfluous
Under this definition, anything that can be omitted from communication due to being already covered elsewhere can be termed a pleonasm.
The etymological argument, for what it's worth, can also be brought forward, as Etymonline explains that pleonasm comes from πλεονάζειν, which simply means "to be more than enough" or "to be superfluous", which in turn simply comes from the root for "more" (the same root that generated Latin and English plus, I believe).