Excellent question. Yes, when you trim - you trim for angle of attack, all other factors notwithstanding (i.e., aircraft configuration, location of Center of Gravity, etc.) The fact that trimming an aircraft will hold the same speed with changing power settings is an ancillary consequence of trim in so far as level flight is concerned.
This trimming for angle of attack also holds true for an aircraft within a banking turn. Because of the greater wing loading within the turn, itself, the natural trim of the aircraft will actually seek a higher airspeed to maintain equilibrium at that same angle of attack. If power is not increased, the airplane will descend in search of this equilibrium. However, if power is added, and the airplane is banked to check its ascent, it will continue to hold the same angle of attack but at the higher airspeed.
Incidentally, I've been told this is why the Navy initially started for fly constant angle of attack approaches as opposed to constant airspeed...even before the advent of AOA Indicators and Approach Indexers in the late 1950's.
Back then, they were flying basically a flat approach from the 180 position into "the groove" just astern the ship, dropping to 90 feet above the surface to give themselves just over 20 feet of clearance at the Fantail, not accounting for any pitch and heave introduced by a ship steaming into the swells, before an LSO issued a "cut-signal" whereupon the pilot would chop power, hold the nose, and plop down onto the deck. But this precariously low altitude became outright dangerous in the jet age with approach speeds of 120 knots. To forestall the specter of flying into the waves approach procedures were codified to insist that, by the time a pilot reached the 180 position, they be configured (hook, wheels, flaps - down), and trimmed-out "on-speed" before commencing their turn into the ship's wake. Once done, power was to be added to maintain altitude within the turn but NO PITCH COMMANDS WERE TO BE INTRODUCED! They were to accept whatever speed this trimmed condition demanded. Starting at 120 knots, and using a quarter-banking turn (22.5 degrees), this resulted in an airspeed increase of around 5 knots. Apparently, it worked and has been gospel since.
Whether this story is apocryphal or not is a matter of debate - I got it from a retired Navy Commander whose career spanned from the late 1940's into the 1970's; from flying props to flying F-8 Crusader's. I've never been able to find any published documentation to support this, but it seems plausible enough to share it with you here today.