"Maintenance" comes to us basically unchanged from Old French "maintenance," meaning upkeep, shelter, or protection—which itself traces back to the verb "maintenir," "to keep, sustain, or persevere in" (the same root behind "maintain"). English borrowers picked up the word in the mid-1300s, originally spelling it "maintenaunce," and the spelling has simply smoothed out over time into the modern form without any major overhaul or Latin-Greek dress-up job—it's stayed close to its French original throughout.
What's more interesting is how the meaning shifted over the centuries. The earliest English use was surprisingly specific and legal: it described a lord or his followers meddling improperly in other people's lawsuits—essentially throwing their weight around in court cases that weren't their business. By the late 1300s, the word had broadened to cover the idea of providing someone with life's necessities, or financial support in general—the sense of "maintaining" a person. Then, in the early 1400s, it picked up the more familiar modern meaning of keeping something in good working order. So the word started in the narrow world of medieval legal shenanigans and gradually widened out to cover both supporting people and maintaining things, which is the sense that's stuck with us today.
maintenance (n.) mid-14c., maintenaunce , "wrongful interference in others' lawsuits by a lord or his followers," from Old French maintenance "upkeep; shelter, protection," from maintenir "to keep, sustain; persevere in" (see maintain ). Meaning "action of upholding or keeping in good order" is from early 15c. That of "action of providing a person with the necessities of life," also "financial provision or support, that which maintains or supports" is from late 14c. also from mid-14c.