There are multiple dan scales. The KGS scale is an amateur dan scale. I don't know how much the scales overlap generally, but I'd imagine a 9 pro-dan professional to be somewhere around 12 dan on amateur scale (pro scales also have more dense scaling). However, the scales reach the ceiling at 9 dan by convention.

Even the abbreviations differ: 9d (amateur dan) vs 9p (pro dan).

KGS 9 dan players are pros or amateurs that are professional level like former insei. The highest rank is almost 11d (it still says 9d, but the graph goes even higher):

https://www.gokgs.com/graphPage.jsp?user=leqi

There's an Elo rating table on this Wikipedia page which pretty much corroborates/agrees with this:

https://en.wikipedia.org/wiki/Go_ranks_and_ratings

I think those servers have their own ranking system that does not match the "official" rankings (which I think cap amateurs at some lower dan rank)

This is correct. A players rank will typically differ between both server and go association. Sensei's Library holds more information: http://senseis.xmp.net/?RankWorldwideComparison