The in-universe reason is well covered by another answer.
Is there an out-of-universe explanation? I can't think of a story reason the Asari's natural abilities need to use the same phenomenon as Reaper tech.
Because Mass Effect is harder sci-fi than you might be otherwise accustomed to.
In Star Trek, the various bits and pieces of technology are only loosely connected to one another. There is a vague sense in which (for example) the warp engines are "related to" the subspace radio. But this is never really elaborated on, and doesn't apply to every pair of technologies (e.g. the warp drive has nothing to do with the universal translator, as far as I can tell). The simultaneous imagining of so many different categories of technological and scientific advancement is what places Star Trek on the softer side of the sci-fi hardness scale.
Mass Effect is different. It quite intentionally limits its speculative fiction to a single change in the laws of physics (the titular mass effect). As a result, there is a far more robust relationship between all of the different pieces of technology. Biotics and ("conventional") FTL, for example, both "run on" the mass effect, as do kinetic barriers, mass relays, and so on. This allows the writers to spend more time developing the fictional physics of the Mass Effect universe, while also making it feel more real by being closer to real-world physics.
Admittedly, Mass Effect 3 somewhat compromises this purity, by postulating that the Reapers have a far more advanced understanding of AI than is typical among the other races. This is not entirely unreasonable, given that the other races systematically avoided developing AI technology, but still problematic, as the geth ought to have made some progress in their 300 years of exile. The use of quantum entanglement communication is also somewhat at odds with reality, but this is quite isolated from the rest of the fiction; we never see quantum entanglement used in other "clever" or unorthodox ways.
Despite this "rulebreaking," Mass Effect is still broadly a work of hard sci-fi by modern standards (although it is quite soft by classic sci-fi standards). As such, it would violate genre norms to introduce additional scientific explanations for the mass relays or for the asari's biotic abilities.