So you booked with miles which means it's not a codeshare unless it's a regional flight. All the mainline flights are booked on the operating airline's own flight numbers. So if your flight number is KLxxx you're flying KLM. If it's DLxxx you're flying Delta.

A codeshare is only when the flight is booked as KLxxxx and operated by Delta as DLyyy. And codeshares are awful You almost never have to book them, and shouldn't unless the price is lower. You can almost always book the operating airline's flight directly as a muti-carrier itinerary on the same ticket if you book via an online (or real) travel agency such as Expedia, Orbitz, or better yet, Chase Ultimate Rewards Travel.

But you are absolutely right, using FlyingBlue you can book one-way awards on Delta and even Alaska Airlines, including domestic flights within the US (if you can find them haha) for half price.