Are there 5000+ companies with IATA (two-character) codes?
It's obviously an error in that Wikipedia page. The "5000 airlines with IATA codes" text actually links to:

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

which lists lots of airlines (over 6000), but most of those have ICAO (3 letter) codes, but not IATA (2 letter) codes.