I'm going to refer back to my answer here.
The original Ymir exists inside of the Coordinate as a thinking entity. She is the one who is the gatekeeper to who is allowed to use the power of the Founding Titan. Due to her time as a slave and all the trauma she suffered in life, she was very loyal to King Fritz and did whatever he told her to do. This loyalty can be thought of as mental conditioning, brainwashing, or a fear of disobeying, but whatever the case, it was shown as a fact in the episode From You, 2000 Years Ago that she obeyed Fritz unconditionally.
This "loyalty" continued even after Ymir died and she was reduced to whatever she is inside the Coordinate. According to the wiki at least, it does indeed seem to be stated somewhere that everyone who can turn into a Titan is descended from Ymir. So, what's the distinction between those of royal blood and the regular Subjects of Ymir? We don't actually have to understand the distinction. All we have to know is that Ymir knows the difference. Perhaps the distinction is even nonsensical, but so long as Ymir believes there is a distinction, she will use it to decide whether to obey or not. She exists as a thinking entity, after all, and with her mind as broken as it is, it makes sense that it wouldn't make sense. Once you put a person in charge of deciding these things, rather than a force of nature which cannot err, all bets are off.
And this is just speculation now, but perhaps, as with the vow to renounce war, the choice for Ymir only to obey members of a certain lineage was something imposed on her some point after King Fritz. Or perhaps it was a sanity check for Ymir, because with so many descendants, she'd almost have to give precedence to some (though this particular theory might not hold much credence, given how time works vastly differently inside the Coordinate). There's been some suggestion that inbreeding of the royals may also have kept the royals more genetically similar to Ymir, which could be the case. Whatever the case, if the explanation makes sense to you, it might make sense to Ymir, and Ymir's the one deciding, so...
Getting back to my original theory, what exactly is the proof that Ymir is the one deciding who is royal and who isn't? It's the fact that she's able to change the rules!
In the same episode, Eren Yeager is able to gain control of the Founding Titan. Sure, he has to play by the rules to get to the Coordinate and get an audience with Ymir, mainly by going through Zeke who has royal blood, but once there, he's actually able to convince Ymir to let him use the power. I forget exactly what Eren said to convince her (might check later and update this), but essentially he appeals to Ymir's desire to stop being a slave.