Not using anything that remotely resembles genetics as we know it.
What you are talking about is fairly similar to the Asari from Mass Effect, who likewise use the genes of other species to "modify" their own. The problem with this is that even if there is a mechanism for combining DNA arbitrarily, you can't plug bits of DNA into another species with a different evolutionary path and expect to come up with something meaningful. Even supposing they happen to look like humans, they will be completely different on a chemical level. They will probably not be able to produce proteins from our DNA, and if they can, said proteins will most likely be either useless or toxic.
The reason why mating makes sense biologically is because you're combining your DNA with that of another organism whose genes are close enough to your own that the child will be able to benefit from those good (or at least viable) genes. Here, the child will not benefit from the fact that the other species has good (or at least viable) genes, so what's the point? You could say it's just randomly mutating to prevent the child from being a clone, but in that case you might as well just randomly mix up DNA and hope something comes of it, saving the tedious difficulty of mating.
Some bacteria do swap genes randomly with bacteria of other "species", but this only makes sense because #1 bacteria are much more structurally simple than multicellular organisms and therefore producing a new protein has a much lower chance of simply killing them and a fairly decent chance of providing some benefit, and #2 bacteria reproduce so fast that they can afford to take these risks.
All this is without even getting into the issues of the species being alien. They might not even have DNA.
Now of course since these organisms are both alien and artificial you could say that they have a completely different chemistry. Maybe they even have "smart" DNA that is somehow able to analyze the DNA of its partner and pick out bits that might be useful. But this is getting into realms of bio-engineering so far beyond feasible speculation that it might as well be magic. Not that there's anything wrong with that.
One interesting solution that is more well thought-out than most is presented in the webcomic Schlock Mercenary. In it there is an intelligent, amorphous species that reproduces through binary fission, and passes a part of its actual brain to the child. They are capable of "breeding" with other species, not biologically or chemically, but by using their own idea of the other individual's personality as a template to build the brain passed to the child, so the child winds up with some character traits of the other "parent". In other words, they mentally construct the traits of their child and use another organism as "inspiration".
This is fairly plausible even for non-amorphous creatures; after all the mind affects the chemistry of the body and the chemistry affects the development of a child, so it is possible for the mind (at least the subconscious, emotional parts) to affect the development of a child. Humans have a relatively small number of chemicals that regulate emotion, but if this species has a much more complex mind-body chemical connection, their thought processes might be able to have significant impact on the traits the (technically cloned) child develops.
This kind of "cross-breeding" by it's very nature doesn't need to involve any kind of traditional mating or indeed any physical contact at all, although you could make a version of it where a physical act is necessary for "initiating" the process. Perhaps they naturally do this to some extent when mating with their own species, and are also biologically capable of cloning, but if they clone themselves while in the presence of another sapient being they can mentally pass traits of the other organism to their child.