Decomposing The Question
There are really two questions buried latently in your question.
One is the (in principle) historically testable (and debatable) hypothesis that the intellectual source of modern identity politics in the United States can be traced to the migration of Critical Theory scholars from Germany to U.S. universities in connection with or after World War II.
The other is the impact the ideologies and tactics associated with "identity politics" has, or can have, on free speech, which is pretty much directly observable today.
Is Critical Theory The Source Of Modern Identity Politics?
Critical theory is a term originally used to describe a school of thought particularly associated with academics in Frankfort, Germany, which is rooted in Marxist philosophical ideas, and is seen by many (especially many conservative intellectuals) as the root source concepts in most of the identity oriented scholarship done in academia and in turn identity politics. In this sense it is the granddaddy of a significant share of modern feminist, anti-racist, and LGBQT scholarship and thinking (although certainly not all of it).
Also, in a broader sense "critical theory" is often used as a general term to describe the shared approaches used by many identity oriented scholars in an ahistorical way that doesn't necessarily imply a true and direct intellectual ancestry from this small but influential group of German scholars.
Others see the source of the shared root ideas of identity politics not in German Critical theory, but in French post-modernism, but the common thread of people using either as a rhetorical device, is that identity politics has a common, mostly European intellectual history that migrated to and took root in American academe sometime in connection with or after World War II and gave birth to a radically different perspective on the appropriate character of academic discourse.
The premise that the modern identity politics movement is derived from a single group of scholars is not well supported empirically. My observation from reading class syllabi and talking to people who are active in identity politics scholarship during my academic life, is that leading academics in those fields read German Critical Theorists, French Post-Modernists, anti-imperialist/anti-colonial scholarship, Marxist scholarship, and lots of home grown scholarship and scholarship from the Third World, in addition to French and German scholarship, in a manner that contributes to a shared and significant philosophical movement mostly rooted in the academic left in the humanities and social sciences, with many parents who have combined their ideas in a classic American melting pot.
Critics of the hypothesis that Critical Theory is behind anti-free speech tendencies on the left wing of American politics, see the links between modern identity politics and Critical theorists who migrated to the U.S. from Germany, as an unfounded conspiracy theory. Critics of this hypothesis also note that often people who see Critical Theory as foundational to modern identity politics also see those Germany academic migrants as Jews in an anti-semitic twist on this hypothesis that Critical Theory is undermining free speech in the United States.
How Can Identity Politics Impact Free Speech?
Identity politics (arguably) tends to devalue statements about identity issues made by people who don't share the identity in question, and to also see the harm caused by statements that are discriminatory or offensive for reasons related to identity to be tangible harms that should be subject to punishment as opposed to the classical liberal stance embraced by the First Amendment that "words can never hurt me." See, for example, a discussion here, which notes in a discussion of identity politics and free speech that:
In fact, censorship has come to be seen as more than the norm. For
many, censorship has come to be a progressive act, a means of
protecting people, challenging power. Restrictions on hate speech,
they argue, protect those facing racism or homophobia or misogyny.
Restrictions on offensive speech protect the dignity of powerless
groups. The use of trigger warnings protect the emotionally
vulnerable. And so on.
Identity politics (arguably) often views rules relating to decorum, civility, and the like in public discussions to be "tools of oppression" by those who hold power, rather than as neutral facilitators of discussion, and similarly, tend to be skeptical that regulations of speech via something like the U.S. First Amendment can ever really be viewpoint neutral. Identity politics analysis, like Marxist analysis before it, tends to be consequentialist, in the sense of analyzing every rule and situation in terms of who wins and loses from it, rather than using extrinsic process oriented notion of fairness.
Paradoxically, identity politics can also focus on the offensiveness of particularly, sometimes commonly used terms to describe something related to an identity (e.g. "Derpy" in reference to someone with a developmental disability, or "Indian" in reference to a Native American), and it adherents can try to enforce abandonment of the allegedly offensive terminology on the grounds that it is offensive or impolite.
Generally, within an identity politics analysis, the norm of preventing offense, or discrimination, or establishment conspiracies from harming members of an "underdog" identity trumps the value of freedom of speech, while traditional academic norms have taken the opposite position.
In application these ideas (arguably having their source in the Critical Theory movement) can be used by a group of people who adhere to these concepts to rhetorically shut down a speaker whom they think lacks the identity standing to make legitimate statements about something related to identity, and can justify for members of that group literally shutting down that speaker by preventing the speaker from speaking on the grounds that the speaker's statements are harmful because they are discriminatory, offense, or that they otherwise cause concrete injury, even if that speech wouldn't offend traditional norms of academic discourse and academic freedom.
