Or to offer it under a commercial licence in parallel.

While that is the most common use case for CLAs, it is normally done by contributors granting a very permissive, but not exclusive, license to a legal entity like a company or foundation, in addition to the public license granted to everyone.

This is not that. This is not even a license. They want a full transfer of intellectual property ownership. Sure that enables them to use it in a commercial product, but it also enables them to sue if contributors contribute similarly to other projects. Obviously that would create a shit storm, and there is an exception with the public license, but riddle me this: can you legally make similar contributions to multiple projects that have this type of CLA?

Let us take a step back and instead look where such terms are more common: employment contracts.

That doesn't require full copyright assignment, though, right?

Obviously this is not the only reason - even the Free Software Foundation require IP assignment via a CLA.

Whether you can or will sign one is a different matter (I will not).