To be fair, this isn't restricted to Mass Effect.
You're not going to include the Captain (Kirk) or the Executive Officer (Riker) in most away teams: Star Trek.
You're not going to have your pilots (Kara) and your CAG (Lee) serving as security when you have marines: Battlestar Galatica.
You're not going to have your Generals as part of an infiltration team (Solo) or flying frontline (Calrissian) in the attack: Star Wars.
For that matter, almost all science fiction gets the role of the Executive Officer wrong and the definition of the ship class Dreadnought wrong.
Specifically to Mass Effect, however, it doesn't appear as though the Alliance actually has anything similar to a standing army in today's world. It's very possible that the word "soldier" has been co-opted to mean non-flight crew in a military where almost everything is based partially or fully on spacecraft. Especially in a military that clearly defines their roles in six neat categories (Soldier, Infiltrator, Engineer, Sentinel, Adept, Vanguard) based on their tech and biotic prowess. Which doesn't explain why an Engineer Shepard would refer to herself as a Soldier, but the true definitions of words rarely retain strict adherence.
This isn't the first time this kind of thing happened.
For example, ultimate does not mean "the best," or "non above," or "will never be topped," or any of the other colorful misuses that have been commonplace. It means "final" or "the last." The ultimate match of a tournament is the final match, the penultimate match of a tournament is the one prior to the final, etc.
Or, more recently, impact. Impact is a noun. It described the event of things colliding into each other. It is not a verb. An event can have great impact, but an event doesn't impact the status quo. We never have things that are impacting.
But through common misuse and adoption, these words have taken on alternate meanings beyond the scope of their original usage.