Not if the discs are mass manufactured. To manufacture them, they produce a master image containing the content, make a glass master from that, and use that to press large numbers of identical copies. To have individually watermarked streams would require making a unique master for each copy.

The best they could theoretically do is to have a write-once area on each disc that has a serial number inscribed onto it in the factory, and have the Blu-Ray spec require that all players read this and then add a watermark to the video they just decrypted from the mass-manufactured disc before passing it on to be displayed, though that would be more fragile, and soon enough noncompliant players would appear, and it would be common knowledge that this exists.

As for why they don’t write the whole disc the way they could write a serial number, that’s much like why they don’t make the whole airplane out of black-box flight recorder material: it would take a lot more time (both in writing and in creating a unique image for each copy) and be a lot more expensive. I understand that they do this for screeners sent out to Academy Awards jury members (which would be just on artisanally burned BD-R discs), though it wouldn’t scale to the mass market.