From what Sheepy has said on stream, it seems to just be a pseudo random number generator picking numbers so no. There is no significant difference between choosing all the same numbers or choosing different ones each time.
I would like to add as a member of BK who was involved in the discussions to decide which name to endorse: Bigger Blacker Knight is a reference to Card Against Humanity's Bigger Blacker Box.
A small quality of life improvement could be letting alliances see which of their members are voting for a bloc name and for which name they're voting for.
Without a profit motive, how would people decide what they should work on? Why should they work on curing cancer rather than making a machine that counts blades of grass?
I'm surprised this hasn't been brought up before given how many times it has been suggested. I'm with Sakura though, I'm doubtful as to whether or not it'd add much beyond confusing micro leaders but I'm not totally against the option.
I've lost count as to how many times people have attempted to form such a thing. To your credit though, I don't recall any of them writing so much for it.