One sort of “rule” I’m working with in my world is that among carbon-based, protein-having species, even those from different planets with no shared evolutionary ancestry, proteins that may be composed of different analogous sets of amino acids still fold into the same overall shapes, paralleling each other in structure when they have the same function (especially since shape often decides function for a protein) regardless of their specific biochemical makeup. In short, protein shapes that perform the same function are highly conserved across species, even those not related. What sort of explanation might suffice to cover why this degree of conservation would occur for reasons other than a shared evolutionary history? Is it enough to say that “same function = same shape” even across different planets like that?
Nothing. There is nothing that would cause protein shapes to be the same across urelated species.
Even here on Earth you can have completely different proteins with completely different shapes having the same function. True, there can be similarities in form when doing the same job with the same mechanism, but biology can often do same thing with different mechanisms. And it is higly unlikely a different biochemistry with different amino-acid analogues will use exactly same mechanisms to do something. Even if the environment would be identical to Earth. And every change in the environment increase the odds of divergence.
What would cause proteins to be the same shape? The laws of chemistry and the same goal. Protein shapes are the function. In natural biology shapes are conserved more than actual amino acid sequences. But when other planets get involved there may be other shapes that work good enough. Point being evolving something that will work a little bit probably isn't as near impossible as some would say - and when something works a little evolution can rapidly make it work a lot better. Also proteins are modular (called motifs).
I'd personally speculate things like alpha helixes will show up on any planet using amino acids.
Something like molecular chlorophyll will probably use the same basic chemistry and have a little variety. But a peridinin-chlorophyll-protein complex will have a lot of alternate solutions with similar functions in other biologies.
We are at the dawn of computers being able to predict structures accurately. For example this experiment used AI to design a protein with a goal function - and the sequence was quite different than known natural sequences.
Realistic yet novel designs
"They tested their models by comparing the new proteins to known proteins that have similar structural properties. Many had some overlap with existing amino acid sequences, about 50 to 60 percent in most cases, but also some entirely new sequences. The level of similarity suggests that many of the generated proteins are synthesizable, Buehler adds."
A Progenitor Race... You don't want any Shared Evolutionary History, so let's throw out the evolutionary part of all planet's shared history with respect to intelligent life. Maybe in your universe, there was a Progenitor Race which visited all of the planets where intelligent life now flourishes. Maybe those genetic engineer's seeded life across the galaxy, building it a little different each time, but always adhering to some basic design elements.
Why are so many species built of similarly shaped proteins? Because those are the shapes which the Progenitors knew would work.
Why are those shapes implemented differently on each different planet? Because doing things the same way every time is boring and the Progenitors enjoyed the challenge of diversity for artistic reasons.
They also could have been changing the recipe on every planet because another ancient race (the anti-progenitors) were launching progressively more advanced pathogens at the each planet they successfully seeded, basing each bio-attack on samples taken from the previous planet. Each planet had to be different or the anti-progenitors' current viral arsenal would wipe out everything the progenitors had planted.