They do come apart, but it looks like the 'tongue' in yours has broken. I had a few hours of fun freeing up mine recently - gone back to stock links and polly from aftermarket links with bearings in
I think that tongue is there to stop the link twisting against the bushes when doing alignment, keeps both ends straight/inline when there assembled and you turn the adjuster bolt.
May not be the worst thing in the world if the link is all free and whoever does the alignment knows about it - could get around it by using mole grips to stop it twisting?