It's a rivalry in the same way that Kevin Keegan and Alex Ferguson was a rivalry.
Not that it was during Pep's tenure but Ranieri has the same amount of premier league's as Klopp with a far inferior side.
Klopp is a good manager but gets way too much credit in my opinion. As you say, it's way too one sided to be a true rivalry.
Mourinho, Wenger, Ancelotti, Ranieri, Mancini, Conte have all won more or the same premier league's as him and its not like they weren't coming up against very good sides with money and managers. Yet most don't get anywhere near the bumlicking he does.
Yeah he won the champions league once too, but it's not like they were the best team around, same with Chelsea in 2021. The best team in Europe doesn't always win the knockout competition. That's why prefer the league as a measuring stick in these matters.
I won't sit here and say he's not a good manager, I just think he's way overhyped. Not on Pep's level and not even on the level of many of those listed above.
Like him or not, prime Mourinho pisses on Klopp as a manager, as does Wenger when he was the first to really stop the rag dominance of the Premier League in the late 90s, early 00s