No, each banner uses its own pity counter in Genshin Impact. For example, the permanent banner can be pulled on 89 times and give a guaranteed pity 5 star for the next pull, but this would have no effect if you pulled on the promotional weapons or promotional character banners as they have separate counts. Fortunately, the promotional banners do carry over each month to different promotions. For example, I pulled 72 times between the Venti and Klee banners before getting a 5 star, which meant I entered the Tartaglia banner with my pity counter already at 72.
Incidentally, data miners have confirmed that the rate actually improves after 70 pulls, reaching roughly a 20% chance after 75 and jumps up to the 100% on pull 90 if you didn’t get one before hand. I actually got my Tartaglia on my 76th pull, after only 4 single wishes.