I would rule that damage from shadow blade is still damage from a spell
Lets start with the quotes of shadow blade and Aura of Warding:
You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60). In addition, when you use the sword to attack a target that is in dim light or darkness, you make the attack roll with advantage.
If you drop the weapon or throw it, it dissipates at the end of the turn. Thereafter, while the spell persists, you can use a bonus action to cause the sword to reappear in your hand. [...]
[...] You and friendly creatures within 10 feet of you have resistance to damage from spells. [...]
I have quoted all of the shadow blade spell primarily to demonstrate that it does a lot of things that a normal weapon would not. It is magical, you just are proficient with it, it deals psychic damage, it has advantage in dim light or darkness, it dissipates if dropped, and it can be summoned back to your hand. From these we can certainly establish that it has numerous benefits exclusively because it is from a spell.
Because it does all these things, I would consider damage from a shadow blade to still be damage from a spell, especially because the actual mechanics that determine how much damage is dealt are part of the spell description. I would contrast this with using a spell like creation to make a mundane weapon whose statistics would be found outside the spell description and would thus be wholly unrelated to the spell. For me, swords made with creation do not count as damage from a spell.
Because shadow blade is from a spell, and specifically because its exact effects are described within a spell, I would consider it to be damage from a spell and thus be reduced by Aura of Warding.
And now let's talk semantics
There are some semantics to unpack here. There exist numerous summoning spells, but in particular, there are the new ones from Tasha's where the mechanics of the summons (their stats) are defined within the spell description itself. I personally wouldn't consider damage from a summoned creature to be damage from a spell (similar to how attacks from summoned creatures are not magical), but this gets into the world of how many levels of abstraction away do we have to get before we consider something to no longer be damage from the spell.
What if a spell shoves somebody off a cliff? Or some sort of mind control results in the controlled creature taking damage? Or a summoned creature makes an attack? Or a sword made through creation is used? Or a shadow blade is used? It's all this semantic mess, so perhaps at the end of the day the real answer is that it's up to the GM.