FIRE is a relatively new free speech advocacy organization started after the ACLU abandoned its very principled stance on free speech rights, which it had adopted in the 1970s, mostly famously in the Skokie Case where it supported a neo-Nazi organization.
Now, many of the organizers and funders of FIRE clearly have conservative leanings. But as far as I can tell, so far FIRE's advocacy has been just as principled and non-partisan as the ACLU once was. See, for example, FIRE pushing back against de Santis' retaliation against Disney: https://www.thefire.org/news/blogs/ronald-kl-collins-first-a... Or this recent case opposing a school board's overly broad censorship of sexually explicit material in its libraries: https://www.thefire.org/cases/penncrest-school-district-bans...
But, yes, by appearances many of their high-profile cases involve liberal colleges, but that's because of the strain of pro-censorship ideology which has become prevalent in liberal circles, causing a dramatic swing in the left's attitudes regarding free speech. Today, advocacy for expansive free speech rights has largely fallen into the hands of a mishmash of cross-partisan people and organizations, more so than in the past, where support on the left had once been more prevalent. And funding, specifically, now seems to be coming more from the right. But if once upon a time the ACLU, whose membership and funding predominately came from the left--even decades ago, not to mention today--could do it, then there's no reason why FIRE can't.
OTOH, the ACLU did a 180 on free speech because over time too much of its membership and, especially, funding came from the left.[1] And the left's attitudes wrt free speech has changed. So hopefully FIRE will make it a priority to expand its funding and membership bases, so it's less susceptible to political shifts.
[1] This isn't a rant against the ACLU. They still do alot of good work. The ACLU has been having these debates in the public for years, and plenty of long-time ACLU supporters have become disenchanted with changes at the ACLU, including previous ACLU leaders, particularly wrt free speech. The reasons for changes at the ACLU are well known and not seriously disputed. The only disputes concern the philosophical justifications for the changes in policy.