No. There are plenty of non-Trump supporters[1] who don't care for YouTube censorship. However, the calls for censorship and pressure on social media companies came as a result of the panic-induced from Trump's election and Russian election interference, which does correlate with anti-Trump voices, more specifically Democratic party establishment and their supporters.

Those aren't the only voices to call for censorship. There is also a parallel cultural war that pressures social media companies to censor hate-speech (as defined by supporters). This tends to generate a lot of noise from both sides, but I think it's probably minor in the grand scheme of things. That Democrats in Congress are cracking down on social media is by far a much bigger deal - because they can actually punish these companies through laws and regulation - and they have been threatening to do so.

[1] I supported Hillary in 2016 because, for all her faults and unlikability, she would have been a reasonable and competent leader. So no, I'm not a huge fan of Trump, but I also don't pearl-clutch every time he says something dumb on Twitter or in a press conference. In a similar vein, though I think he should be a one-term president, I don't think it's an existential threat if he gets elected again.