Is it "weird" that all disc galaxies rotate once every billion years?
It is not weird.

The comment is about a certain class of galaxy, and not just a class but a stage of galaxy formation and development within that class.

We start with a fairly constant distribution of matter within the universe.

Stars for when this matter clumps together. This clumping is going to have a fairly even distribution.

Stars then clump together to form structures that we identify as galaxies.

The distribution of this clumping is also going to have a fairly constant distribution.

This carries on up the chain to clusters of galaxies, and clusters of cluster and so on.

Going back to stars, we identify main sequence stars. These are stars that start of with a mass and composition that is very common and then have common life cycles. They take so long to burn up certain percentages of their hydrogen, so long burning combinations of hydrogen and helium and then so long going up the periodic table forming lighter elements and in the process becoming the different types of stars we define as being on the main sequence.

For galaxies it is similar.

Spiral galaxies need to be of a certain size and angular momentum to generate their signature spiral structure. They did not start out as spiral galaxies. And they will not remain spiral galaxies. They are simply how galaxies of a certain size and angular moments appear at certain stages in their development.

This means that the outer edge angular velocities of these types of galaxies (for a specific definition of their outer edge) are going to be similar.  

For some information on the development of spiral galaxies and other galaxy types:

https://astronomy.swin.edu.au/cosmos/S/Spiral+Galaxy+Formation