Counter attacking was still present before Mourinho (United, as an example or Liverpool). His genius was combining those with a 3 man midfield (with a pure DM role in Makelele) which made his first Chelsea team an absolute domination machine. Clubs in PL just didn't know how to react to the extra man advantage in midfield - especially since most of them were used to a 4-4-2 or a 4-4-1-1. (and none of them were fluid) ****, Ferguson was the first to experiment with a 4-5-1 (during the treble season) and teams thought he was mad.
Mourinho set up his teams to counter-attack, yes, but his x-factor revolved around the 3 man midfield and his setting the team up basis opposition (to an unmatched level of detail) meant more often than not, teams had utterly no response, and that ridiculous record Chelsea set during that time. He created an entirely new role (which was owned by Makelele), so much so that other teams started mimicking it in a bad fashion, converting attacking playmakers into defensive midfielders (Lucas Leiva, Veron etc).
Funny thing is, the Mourinho before RM actually attacked too. Yes, it was counter-attacking setup, but he also had his teams play flowing football lots of times! I think something broke when he went and finished his spell at RM. Became less tolerant, was a lot less charming and his tactics became extremely reactive. I think RM spell was what changed him.