Are there proteins with permanently bound magnesium? Do you know something about the existence of a magnesium containing protein/enzyme in which magnesium is coordinated in the structure and the structure is stable? (like iron in haemoglobin or zinc in zinc fingers)
From the look of things you may actually be referring to a lot of different proteins, however I managed to
grab two that I found fascinating:


Enolase


A glycolytic enzyme which catalyzes the reversible dehydration of 2-phosphoglycerate to phosphoenolpyruvate:

 

Credits: Lehninger Principles of Biochemistry page 213 ("Enzymes")


In the enzyme active site, 2-phosphoglycerate undergoes strong ionic interactions with two bound Mg2+ ions causing the resulting intermediate to be easily abstracted.





GTPase specifically  ARF1 (ARF for ADP-ribosylation factor). These are important in priming phase in the formation of Clathrin-coated vesicles (CCVs).


Clathrin itself is  a protein that forms a polyhedral framework around vesicles that transport TM, GPIlinked, and secreted proteins from the Golgi to the plasma
membrane.



X-ray structures of ARF1.GDP and ARF1.GTP, were determined by Dagmar Ringe and by Jonathan Goldberg:



Credits Voet and Voet Biochemistry 4th ed page 435 ("Membrane Assembly and Protein Targeting")

X-ray structures of (a) ARF1.GDP (left) and (b)ARF1.GMPPNP (right) (GMPPNP is a nonhydrolyzable GTP analog in which the O atom linking GTP’s beta- and gamma-phosphorus atoms is replaced by an NH group.) 

The bound nucleotides are drawn in stick form in white with their phosphorus atoms magenta and their bound $\ce{Mg^2+}$ ions shown as lavender spheres.

In ARF1.GDP, the protein’s N-terminal helix (red) together with its
covalently linked myristoyl group (not present in the X-ray structures) are bound in a shallow hydrophobic groove on the
surface of the protein formed in part by the residues of loop gamma 3.