First, as TukishVan and Tryst basically pointed out, the stallion does not "lose" it's registration. It doesn't suddenly stop being a pure blooded Friesian stallion. If it has made it to the breeding stud book in the Dutch registry it will loose it's position and status there and be dropped from the list of stallions approved for breeding.
Relatively speaking very few stallions get there. Most that make become STER don't. Not because they can't, but it is a lot of work, time and money
Now, that's for the Dutch registry. The requirements are not the same outside of Holland. Different registries exist in other countries and they can each have their own requirements which may or may not be the same (e.g. The FPVZ in Germany differs from the Dutch FPS, and there must be at least 3 different registries in the US. Some of which will certainly differ from the FPS). Some don't allow it for approved breeding stallions (e.g. The FPS), but most do allow cross breeding.
As DraftyAriesMum comment about. Yes, todays Friesian is a breed that had to be basically brought back from the brink. They did this by cross breeding (oh parish the thought
) and not so very long ago really. They used breeds that had used Friesians in their development. Basically the same as if it had to be done today they might use the Friesian Sporthorse to help bring back the breed.
The Friesian is (and historically as been) a great breed for improving and developing breeds. In much the same way the Arabian has been used, but for different traits. (calm, strong bones, etc..)
For those who think it's a bad idea to cross just keep in mind that ALL of the breeds we have today (all the way back to first "recorded" horses used my man for which we have documented records) were the product of (selective and non selective) cross breeding. Sometimes we end up with superior horses and sometimes not. Ultimately time sorts out the breeds that we like to keep around (for whatever reason).
So when you go out and look at that horse (or those horses) that you love so much keep in mind that that animal is the product of someone, somewhere at some time breeding a stallion from one breed (or perhaps a cross breed) to a mare from another breed (or perhaps some other cross breed) and over time, no matter what the breed, the end product is what you're riding today.