Most of the stories being told are about events happening not just in alternate universes, but on alternate Earths.
- The early stories involving (for example) the Squadron Supreme, or the alternate Earth where Reed Richards became the Thing, were all indeed set on Earth, not some other world.
- Most (certainly not all, but most) of the stories in What If took place on Earth.
- Age of Apocalypse? Primarily (if not exclusively) set on Earth.
- The various Marvel TV shows and animated series predating the MCU - again, mostly set on Earth.
- Even with the MCU, especially taking the Netflix shows into account, primarily set on Earth (the Thor and Guardians of the Galaxy movies being notable exceptions).
It's also true (as noted in another answer) that DC had been using Earth-whatever (1, 2, 3, S, X, 4, C, C-) since the early 1960s, making that a natural nomenclature for Marvel to pick up.
Noting that DC originally used the terminology, and that DC had more than one SF writer in the early 1960s, one might be surprised that they wouldn't have gone less local with the terminology. However, again, most of the Earth-1/Earth-2 stories took place on Earth.
One could suppose a "Chekhov's gun" issue being involved. If you refer to Universe-1, the reader is likely to expect the story to not take place entirely on Earth.
A final note: I'm not sure when the terms "Marvel Universe" and "DC Universe" became common. Honestly, I expect it trickled in from SF fandom, where most if not all works by an author might be seen to take place in the same universe (and where "on the same Earth" would be silly, since few of those works might have been placed on Earth). I honestly can't recall how the separate realities were referenced in, say, the 1970s (and yes, I was collecting comics then). So, using "Universe" instead of "Earth" may simply note have been thought of at the time.