It does not, but your DM might decide otherwise
The text maybe implies magic because the wings are spectral, but it does not say the wings are magical. So by a strict reading of the criterium "Does its description say it is magical" the feature is not magical and will not be suppressed by Antimagic Field.
For example, the cold breath of an white dragon also feels very magical, but is not. The SAC tries to justify this as follows:
You might be thinking, “Dragons seem pretty magical to me.” And yes, they are extraordinary! Their description even says they’re magical. But our game makes a distinction between two types of magic
- the background magic that is part of the D&D multiverse’s physics and the physiology of many D&D creatures
- the concentrated magical energy that is contained in a magic item or channeled to create a spell or other focused magical effect
By that logic, spectral wings are just part of the physiology of the creature, "background magic".
To be honest, I feel this is lame and arbitrary. But at least, there are clear rules to follow to draw the line.
The game's language is not unclear here. It just leads to a result that is not what one would intuitively like it to be.
And that's why we rely on the DM to apply judgement. If a DM does not like this outcome, and decides that "In my game, by gosh, an antimagic field will supress your dainty spectral wings!", the DMG on page 4 gives them the right to rule so:
The D&D rules help you and the other players have
a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game.