I’m more inclined to think alien biochemistries would be so different from one another that they couldn’t affect one another for good or ill, and certainly one could not and most assuredly would not want to eat the other. If speculative silicon life forms existed, for example, I imagine to us they would be like living rocks, and nobody wants to eat a rock.
Even a biochemistry that uses the same chemical elements as ours is likely to be very different. Some of our proteins' amino acids are prebiotic, while others aren't, and the latter ones are likely later add-ons.
The most plausible hypothesis to date for the origin of the DNA - RNA - protein system is the RNA-world hypothesis, which states that RNA was both information carrier and enzyme, "ribozyme". DNA is a modification of RNA that only does primary copies of genetic information, with messenger RNA being secondary copies. Proteins are assembled on ribosomes, RNA-protein complexes with the RNA parts being the primary working parts. So both DNA and sequenced proteins are a result of the RNA world.
About RNA itself, while the nucleobases are mostly prebiotic, the ribose maybe wasn't. It's hard to make sugars with prebiotic chemistry, and when one tries, one gets mixtures of every possible asymmetry of each molecule. While protein amino acids have one asymmetric carbon atom in them, ribose has three.
I've seen speculation that ribose wasn't the first, that it had some predecessor, though it is hard to pin down what that predecessor might be.
If that is the case, then it means that the possible biochemistries of carriers of heredity is broader than in our biota.
More speculatively, I wonder whether their arrow of time would be the same as ours. The block world picture of space time posits that the future exists along with the past, and we know that at the microscopic level processes are time symmetric, with no arrow of time. Could it be possible that an alien intelligence remembers the future? This was the theme of the sci-fi flick Arrival, which left any number of philosophical riddles to ponder include the old standby of free will.
I find that implausible, because the direction of time is embedded in our Universe's macroscopic physics. It's something like the problem of how one gets from quantum to classical mechanics, it seems to me. Both of them are problems that are much more difficult to solve than what they might seem like at first.
I would be willing to bet that most life is toxic in some chemical way to most other life, of life forms of distant planets.
Even life of the same species is subtly toxic in internal ways, generally, for earth.
I sometimes speculate about a lovecraftian organism that engineers originations of life and even fosters enclaves of various kinds on moons, and then repeatedly throwing examples into enclaves of their own form of life in a steady trickle, until the aliens are no longer bothered chemically by those systems of life: a sort of one-way evolution, to be adaptable to every form of life as they can.
They then go out into the universe attached as spores to objects with interstellar velocities, to attach and hatch and survive in the rugged outlands of a system where the gravity is low and energy is meager and time is slow, eventually seeking legrange points and sufficiently large belt objects until their slow biological or electromechanical processes have the accreted material and processing power to make inroads into terrestrial bodies capable of supporting their form of life, which would probably be universally toxic in every way to us, after doing the same for some thousands of years using earth life.
But that's like, cosmic horror level science fiction. It's actually entirely plausible, but unlikely.
Honestly, I expect that the name of this life form is "Earthers" and we will be a terror and a curse on the universe of we ever escape this rock, because this could as well be used as a roadmap in the same way as The Handmaid's Tale.
We exist at the dawn of life in our universe. It's hard to imagine other life getting an earlier start than life around Sol did. We had few mishaps, got plenty of interstellar material from plenty of weird events to make planets and it only took several billion years after the last time a star went "pop" and the gravity well here went a bit *fuzzy*.
Honestly, given how long life bearing planets are going to be precipitated by various stellar events, we're right near "the beginning" even if there were a big rip some time in our perceptual future that isolates us...
The amount of time it takes to evolve whatever various forms of life as various chemical signatures observed planets would imply is immense, especially to a level where such warfare would be possible or even advisable. They would have to appear early and plan the beach-head event.
Engineering how from our level of technology forward to "where it's going" would only take a few hundred years at most without a dark age, perhaps only a few thousand with one or a scant 100m if we managed to apocalypse most life this time around.
It would take a while before we ever encountered a life form that was our equal, and we could put down shop just around anywhere at that point, since we would plan out our game often before other forms of life first precipitated.
But humans aren't known well for their forward planning capabilities.
We're probably going to apocalypse ourselves trying to figure out how to travel whole-human in an interstellar air bottle, and whatever lifeform I described will evolve against the ashes of the life they find here and move in.