I believe Doohan was as vitriolic against Shatner through the '90s and early 2000s so not every supporting cast member but also not super-unique to Takei. And it's likely that part of it was/is wanting to play up the feud for yes attention and convention-drama and -support.
I remember a long, long time ago (probably long enough that it's fallen off the board archives) reading someone here talk about Doohan having a convention anecdote about calling Shatner after hearing they were going to be coming back for a Star Trek movie (presumably, this is before it became a TV show, then became a movie again), and the first thing Shatner said was, "How did you get this number?"* Years later, this person saw Doohan at another convention and he gave the same story, but this time, Shatner was all, "How dare you call me, sir?! You, who are not fit to darken my door! Cease to disturb me, peasant!" (I don't remember the exact phrasing, but the idea is that in the telling, Shatner went from being aloof and disconnected to being enraged at Doohan calling him).
In retrospect, I agree that these feuds were more of a way to grab a bit more limelight with some drama, though there's probably a kernel of truth there. Shatner, in his own telling, isn't shy about admitting to upstaging the supporting cast, on the grounds that if he hadn't jealously guarded his status as lead actor, the producers would've fired him, made Nimoy the captain with a token pay raise since Spock was already so popular, and pocketed the savings, though he doesn't make it sound malicious. Shatner himself leans into the persona, too, which could be the reason his twitter account is still (I assume?) being run by that 4chan weirdo who keeps picking fights with people.
*Which, personally, I think is funny as hell.
