Shouldn't matter? Protein folding is based on the laws of physics after all. If DNA sequences folds differently in different organisms then an external factor is missing.

While the laws of physics remain the same, the folding machinery between species varies to some degree. Protein folding is determined by the unique environment/machinery of a cell. A concrete example is disulphide bonds (S-S, ex cystein-cystein) that require a certain pH to form. The primary pathways of disulphide-bond formation are localized in the endoplasmic reticulum (ER) of eukaryotic cells and the periplasmic space of prokaryotic cells. So two complete different mechanisms to end up with the same bond (protein structure) depending on the organism.

Outside of missing post translational modifications, can you give a concrete example of a protein that is known to fold differently in different species, not counting, say, stuff getting sent to the garbage bin of inclusion bodies due to the stress of overexpression? My understanding (7 years of grad school researching protein folding in the ER) is that outside of some rare corner and disease state cases, folding is pretty much binary event, and if it weren't for most cases the low delta g difference between isoforms would be just as easily overcome over the course of environmental changes in a single individual as "between different species" namely having a deterministic outcome is important for through-time robustness.

??? Unless you jump from eukaryotes to archea these are not real concerns. Most PTM markers are very conserved.

I'd say the jump from eukaryotes to procaryotes is a realistic scenario in recombinant DNA technology.

I have some experience with recombinant yeast and PTMs. Degree of glycosylation actually vary a lot depending on strain used and has a huge effect of protein activity. And of course these PTMs affects the crystal structure.