Lazarus Project.
Think about it, Shepard is KILLED in action, and subsequently revived/rebuilt. We know that biotics revolve around brain implants, tech abilities involve omni-tools, combat abilities involve weapons, and whatever your class there are capabilities brought about by gene therapy/engineering as an N7.
Who's to say that the Lazarus Project didn't effectively make Shepard EVERY class? We know that Shepard's resurrection involved cybernetics and brain restoration, so it's completely possible that Cerberus thought: "Since we're in here, we might as well provide the implants needed to use biotics." I get the feeling that since Shepard can be any class in ME2, it's not that Shepard can use ALL the abilites from ALL the classes, but rather he can only be one at a time.
My personal headcanon of my Shepard is that he's a Vanguard/Sentinel hybrid, i.e. I use a Gibbed save editor to switch between the two classes.
They are both half-biotic, with one using weapons and ammo, the other tech powers. The Vanguard is offensive in biotics: raw power and force. The Sentinel is support in biotics: combo's and DOT. Biotics, regardless of type and application, use amps, so I don't see it as too far a stretch to think that my Shepard can "tune" his amp to suit the mission at hand.
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To answer the question of why a non-biotic Shepard can use a biotic bonus power, my best educated guess is the fact that biotics are different for each biotic individual. I believe that Shepard is a biotic, or at least biotic-capable, and the class/bonus power you choose determines just how much they've tapped into their biotics.
My Vanguard uses Barrier in ME2 (Suicide mission, I am going to survive) and Flare in ME3 (Raw power, to tell the Reapers to get off my lawn!)
- weapons are a supplement, i.e. biotics "break down the door" and weapons clear the room.
My Sentinel uses Warp Ammo in both ME2 and ME3 (tech-minded precision applied to my biotic power to improve my ammunition)
- tech abilities remove any advantage my enemies have, and then biotics deal the damage supplemented by weapons fire.
So for example, a Shepard who is a Soldier who focuses entirely on weapon use, but has a biotic bonus power, would only have that one power because they are entirely devoted and committed to using conventional weaponry. The mental discipline and focus is present, but is simply applied onto something non-biotic.
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Long story short, Shepard can be any class, with any bonus power, because the potential is there. We, the player, simply choose where to tap into it and how much.
ME1 because N7
ME2 because Lazarus Project (immediate)
ME3 because Lazarus Project (the cybernetics and enhancements are still there)