It is a well known fact that Alex is more soft-spoken than (she/her).
Why would "her" be wrong? Why must the sentence end with "she"?
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Mainly on the basis of "grammar rules" more suited to Latin, there are still plenty of pedants willing to tell you accusative pronouns (me/him/her/them) are incorrect in such constructions, and that you should use a nominative pronoun (I/he/she/they).
I suppose the "rationale" for that position is there's a "deleted" verb following the pronoun (more soft-spoken than she is, in OP's example). But you've only to look at the chart in this excellent ELU answer to see that native speakers in general have increasingly abandoned that position, particularly in recent decades. So if you want to sound like a modern native speaker, use the accusative pronoun her here.
TL;DR: I wouldn't go so far as to say she is a "hypercorrection", but it's old-fashioned and stuffy.
This handbook suggests that she/her is a matter of preference in this case:
Generally, the only question about "than" arises when we have to decide whether the word is being used as a conjunction or as a preposition. If it's a preposition (and Merriam-Webster's dictionary provides for this usage), then the word that follows it should be in the object form.
He's taller and somewhat more handsome than me.
Just because you look like him doesn't mean you can play better than him.
Most careful writers, however, will insist that "than" be used as a conjunction; it's as if part of the clause introduced by than has been left out:
He's taller and somewhat more handsome than I [am handsome]. You can play better than he [can play].
In formal, academic text, you should probably use than as a conjunction and follow it with the subject form of a pronoun (where a pronoun is appropriate).
So in your sentence
It is a well known fact that Alex is more soft-spoken than her.
would be a more informal, but valid, way to say it.
It is a well known fact that Alex is more soft-spoken than she.
is the expected way to say it in school work and business writing.
Go by what makes grammatical sense when you substitute the pronoun after "than" in place of the noun with which it is compared. In the preceding example, "... than she." yields
It is a well-known fact that she is more soft-spoken.
while "... than her." yields the incorrect
It is a well-known fact that her is more soft-spoken.
The choice of pronoun can be disambiguating. For example,
Alex talks with her more often than me.
implies "... more often than he talks with me."
Alex talks with her more often than I.
implies "... more often than I talk with her.",
because these are the interpretations that make grammatical sense when you substitute the pronoun after "than" into the preceding statement in one place or the other.
"She" is a pronoun, referring to a person.
"Her" is a possessive adjective, referring to something of or about a person.
Simplifying the original sentence using correct English for both the pronoun and adjective, respectively, we get:
"Alex is more soft-spoken than she."
"Alex is more soft-spoken than her friend."
Now, append the word "is" to the above two sentences:
"Alex is more soft-spoken than she is."
"Alex is more soft-spoken than her friend is."
They still make sense, although neither is considered good writing (they aren't wrong, per se, but they ain't great shakes, either!). ;)
However, if the word "friend" is omitted from the adjective example, then it is clear (hopefully!) that the resulting sentence doesn't work:
"She is more soft-spoken than her is."
Therefore, using the adjective, alone, is not good English.
"Her" is wrong because the sentence is an abbreviation of a longer sentence, and that longer sentence needs "she" not "her."
It is a well-known fact that Alex is more soft-spoken than she
Is really short for:
It is a well-known fact that Alex is more soft-spoken than she is.
The pronoun "she" has to agree with the verb, even though the verb isn't there.
A clearer example could be this:
I can run farther than he.
is short for:
I can run farther than he can run.