I have heard that the oldest existing airline company is DDL - Det Danske Luftfartsselskab (Danish Airline Company). And it is slightly older than KLM.
DDL does, however, not operate airliners, and has not done so for more than fifty years. So it hardly counts.
They own a lot of airliners - or at least they have them registered on their name on the national register, but they are operated by SAS, which is owned by DDL plus DNL (Norway) and ABA (Sweden).
Always keep your number of landings equal to your number of take-offs