There is more to this question than meets the eye. The question is related to the question 'Gravitational' power vs. engine power and this answer will be related to the answer https://aviation.stackexchange.com/a/56040/34686 . That question used the phrase "exactly identical" and so a very precise answer was given. At the end of the current answer we'll look at whether or not these considerations make any PRACTICAL difference for flight in conventional general aviation aircraft.
You say both aircraft are following the same glide path relative to ground. This means the aircraft with a tailwind is following a steeper glide path relative to the airmass within which it is flying. This means that the glide path through the airmass is aimed more steeply down, relative to the horizon, in the case of the aircraft with a tailwind. This means that the aircraft with a tailwind must have a larger value of (drag minus thrust) than the aircraft with a headwind. Since the angle-of-attack of each aircraft is almost identical (more on this below), this means that the aircraft with the tailwind must be flying at a lower power setting than the aircraft with the headwind.
See the related answer https://aviation.stackexchange.com/a/56040/34686. The vector diagrams in that answer can be adapted to the current question simply by replacing the label "drag" with label "drag minus thrust", and by recognizing that in a powered glide, the glide ratio is the ratio of lift to (drag minus thrust) rather than the ratio of lift to drag. The lift, (drag minus thrust), and weight vectors still must form a closed vector triangle, exactly as illustrated in https://aviation.stackexchange.com/a/56040/34686. Meanwhile the ratio of lift to drag is still governed by the angle-of-attack of the wing: every specific angle-of-attack corresponds to a specific lift coefficient, drag coefficient, and lift-to-drag ratio.
Due to the relationships explained and illustrated in the related answer https://aviation.stackexchange.com/a/56040/34686, including the relationship lift = weight times cosine glide angle, the lift vector is SLIGHTLY SMALLER in the case where the lift vector is tilted forward more, i.e. where the flight path through the airmass is aimed more steeply downward. This describes the aircraft flying at the lower power setting-- the aircraft with the tailwind.
If angle-of-attack were the same in both cases, the aircraft with the tailwind would be flying at a slightly lower airspeed-- that is the only way it could be producing less lift, for the same angle-of-attack.
Since you specify that the airspeed is the same in each case, the aircraft with the tailwind must be flying at a slightly lower angle-of-attack than the aircraft with the headwind. This introduces the complication that the L/D ratio will not be exactly the same in both cases-- but the basic ideas explored in the answer https://aviation.stackexchange.com/a/56040/34686 will still apply.
It is very unlikely that a pilot would be able to directly detect this difference in actual practice in most cases. In actual practice, the angle-of-attack would probably seem to be the same in both cases. For example, consider an aircraft with a L/D ratio of 8:1 at idle power at some given angle-of-attack. In still air, the aircraft can glide 8 feet forward from every foot of altitude it loses. The glide angle at this angle-of-attack is equal to arctan (1/8) = 7.1 degrees, and the lift vector is equal to weight * cosine (glide angle) = weight * cosine (arctan (1/8)). This works out to weight * .9923. So there is a less than 1% change in the magnitude of the lift vector between the idle-power case, and the case where aircraft is generating enough power to fly horizontally at that same angle-of-attack. For a given airspeed, only a tiny change in angle-of-attack would be needed to cause this change in the magnitude of the lift vector.
If we are talking about an aircraft with the glide angle of a brick-- like an aircraft with a L/D ratio of 3:1 or 2:1 or less -- then the difference in the magnitude of the lift vector between the power-on and power-off cases would be more significant, regardless of whether it is the airspeed or the angle-of-attack that we are constraining to stay constant.
So the PRACTICAL answer to your question is, that for most things that fit our definition of an "aircraft", there is essentially no difference in angle-of-attack, and in lift coefficient and lift force, required to fly down a given glide path (defined relative to the ground) with a tailwind or headwind, with the same airspeed in both cases. The aircraft with the tailwind, i.e. the aircraft with the lower power setting, is indeed flying at a lower angle-of-attack and generating less lift than the aircraft with the headwind, but the difference is extremely small.
The only practical way a pilot could detect the tiny difference between the magnitude of the lift vectors associated with the headwind and tailwind cases is by being aware of the fact that a change in the glide ratio (with respect to the airmass) always involves a small change in the magnitude of the the lift vector, as well as a much larger change in the magnitude of the (drag minus thrust) vector. When we are descending at either a given angle-of-attack or a given airspeed, if we reduce thrust and increase the (drag minus thrust) vector and degrade the glide ratio and cause the aircraft to glide in a more nose-down pitch attitude, this always goes hand-in-hand with a very slight reduction in the magnitude of the lift vector. If angle-of-attack is constant, then we've decreased the airspeed, and if airspeed is constant, then we've decreased the angle-of-attack. The change in the airspeed or angle-of-attack will typically be extremely small, but the change in the glide ratio through the airmass, and the resulting change in the aircraft's pitch attitude, is in fact a signal that we've slightly reduced the magnitude of the lift vector by slightly decreasing the airspeed, the angle-of-attack, or both.
A final note-- for simplicity, this answer has ignored the effect of propwash over the wings. In truth, the required difference in lift force between the aircraft with the headwind and the aircraft with the tailwind is so small that in many aircraft it could easily be fully accounted for by the higher power setting and increased propwash over the wings on the aircraft with the headwind. Therefore on any aircraft where there is a propwash over the wings, we can't say for certain that the angle-of-attack is slightly lower when descending with a tailwind than with a headwind at some fixed airspeed and some fixed glide ratio over the ground. But we can still say for certain that the lift vector is slightly smaller when descending with a tailwind than with a headwind at some fixed airspeed and some fixed glide ratio over the ground.
Nothing in this answer should be construed as a suggestion that the wings somehow directly "feel" the effects of a headwind or tailwind. The key is that the pilot is forced to choose a different power setting to stay on the glideslope at some given airspeed when flying with a headwind versus a tailwind. We would see exactly the same effects if we explored the effects of changing the power setting while maintaining some given airspeed while flying in a descending glide in still air.
See related answers to related questions:
"'Gravitational' power vs. engine power" -- https://aviation.stackexchange.com/a/56040/34686
"Does lift equal weight in a climb?" --
"What produces Thrust along the line of flight in a glider?" -- https://aviation.stackexchange.com/a/56371/34686