It is a bit silly, but I kinda like it. I'm generally pretty anti-koolaid, but having a shared unique term for yourself and your coworkers seems like a reasonable way to build some positive tribalism, even if it's a stretch linguistically.
The real curse of PagerDuty, though, is calling everything "-duty". Parental leave is BabyDuty, the training department is LearningDuty, the LGBTQ ERG is RainbowDuty, you get the idea.
Hat-tip to our veterans ERG for not going with DutyDuty, although I hope it was in the running at least.
(I mean I also kind of love it, but you know, it's a lot.)