Yes, I do the same. I like to be frugal and get the most advantageous option. (It gives me the illusion that I can win sometimes...)

Usually, I know exactly what flight I want and before booking, I check not only with operating/marketing airlines, but also their national web sites. Also, KLM US does not charge credit card fees -- while European sites do charge a fee for using credit/debit cards (sometimes based on the country of issuance). And then there is a difference given by the exchange rate (and using AMEX usually saves me 5 % compared to VISA/MC). I recently bought tickets for two people, and the difference for the tickets on the same plane among the different websites amounted to US$ 300.

Moreover, I would never buy Delta E class, but buying the same (cheapest) class with AF/KL on the same flight may be 10 % cheaper than DL and also give me Lifetime and Bonus miles as well as MQDs for DL.
When comparing the same flight on AF and KL web, they are mostly the same, but occasionally, I saved a few dollars by making the right choice. (I guess each of them has a certain allotment and when that sells out on any given site, it can still be bought on the other.)

This does not apply to all airlines, but sometimes Google Flights also gives an option to book at different agents (Gotogate, Expedia...) and they sell the same flight for 10-15 % cheaper than the airline. And it works the same way. Go figure...