You assume that alien's inheritance is guided also by DNA. It is very unlikely that on another planet, under very different conditions, life would evolve to use the same rather complicated mechanisms for inheritance: ATCG for DNA, proteins to guide transcription, etc. They are all complicated and co-evolved to work together. It is like a very complicated dance, with chemical signals passing back and forth.
From the point of biology, humans, animals, plants, and fungi are very similar - all are eukaryotes - even on Earth are many organisms (prokaryotes and archea) which are radically different from eukaryotes, (IOW more different than the difference between an animal and a plant) even if they use the same DNA mechanism for inheritance. For instance, many bacteria do not have sex, but are able to exchange plasmids (which contain some DNA info, but not as string like chromosomes of eukaryots, but as rings), directly. Alien world indeed. Interbreeding a plant and animal would be substantially easier than interbreeding with alien non-DNA life-form.
Even in humans, sometimes proteins misfold and cannot function. Or function, but wrong way. Tha's why we have so many genetic diseases. And miscarriages, when embryo is not viable (chemistry is broken in a subtle way). When cell is dividing, different biochemical signals sent to neighboring cells guide development of different organs. Wrong signal will cause errors on development of organs, and embryo could die or misform.
The only way would be if life on both planets was seeded by some elder race to use same genetic mechanism. Even then, after few millions of years of separated evolution, species would not be able to cross-breed naturally. This happens even on Earth. If both species use exactly same biochemistry for genetics, you might be able to artificially inject genes from one species to another. It is a crap shot, and in most cases this infusion will destroy some biochemical signals or another. Gene expression is very delicate process and there is lot what can go wrong and cell is doomed.
So it is your world, and you can postulate that it is so, but such situation would not evolve naturally - only in a Hollywood script.
Also, some viruses - retroviruses - spread by adding they own RNA to cell's. And cells have all kinds of mechanisms to prevent that. So when someone would want to add alien DNA to the cell, cell would fight back - and has millenia of success to count on. Big chunks of human DNA is suspected to be retrovirus genes injected, but disabled, during evolution. It is much more complicated than you imagine.
(I guess my answer not accepted, too much technical details about the real complexity of the problem :-)