I think when they say "KLM flight" on that page (right before the part you quoted), it has to be understood as "KLM-marketed and KLM-operated". Because there is a different subpage in that same section of the site about codeshares, and if you look through that list, it is definitely not the case that you get Flying Blue benefits when you fly non-SkyTeam airlines (with a KL flight number). The different codeshare agreements vary quite a lot (and if the airline gives you anything beyond what is promised here, like Air Baltic apparently does, it is a nice surprise).

https://www.klm.com/travel/nl_en/pla...ners/index.htm