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I think those groove spindles are designed for handles where there is a grub screw in the handle to tighten down on them. The groove (and thinned metal part) then looks like it is designed to flex and return some kind of preload back to the grub screw so it doesn't undo. Usually seen more on exterior doors (at least thats where I've seen them). The cross section is still square though, so it may just be that your handles are loose. Looks like they used a (too) large diameter initial hole before the square was broached. Might have to live with the play, try to find chunkier spindles, shim the spindles out or return them.
Just fitting some new handles to our internal doors and there is play in the action once installed.
On further investigation the opening for the spindle in the handle isn't a normal square shape.
Has anyone seen this before? I'm wondering if it needs this type of spindle instead
https://www.sdslondon.co.uk/product/grooved-spindle-8-x-90-mm.html