Objects from the Animate Objects spell deal piercing or bludgeoning damage depending on the object type. Is this damage counted as magical against creatures with resistance to non-magical damage?
No, the damage is not magical.
From the description of the Animate Objects spell:
Choose up to ten nonmagical objects within range that are not being worn or carried
Let's say the spellcaster animates an ordinary shovel, which can then fly and slam any target. Because this object is nonmagical, there is no reason that the damage should be magical. It is basically as if the caster would throw the shovel to a target.
Also, for example, none of the animated objects from the MM p.19 does magical damage. The Animated Armor and the Rug of Smothering do normal bludgeoning damage and the Flying Sword does normal slashing damage.
The animated objects do magical damage.
The only question here is whether the attack is from a "magical source."
The spell effect is not "instantaneous," hence it is on on-going magical effect (subject to dispel magic.)
a magical attack is an attack delivered by a spell, a magic item, or another magical source.
There is distinction regarding things like Earthquake: the boulders bouncing around are non-magical damage. The boulders also aren't "attacks," (no attack roll). So they would harm the creature.
The descriptions in the MM are for constructs that have a permanent magical effect. For Golems, another construct, "Magic Weapons. The golem's weapon attacks are magical." The Golem is a permanent effect that cannot be dispelled. So it doesn't qualify as a "magical source."
Contrast with Conjure Elemental, which is also concentration:
An elemental of challenge rating 5 or lower appropriate to the area you chose appears in an unoccupied space within 10 feet of it.
An elemental is conjured, but not animated, so you use the stat block from the MM.
The Animated Armor in the MM is technically not a relevant comparison because it is permanent magic, as are other permanent animated magic items.
Mordenkainen's Sword and Spiritual Hammer do force damage and use the wizards attack role, and are not magical weapon attacks, so no comparison there.
Finally, the stat block in the description isn't a proper stat block. It is incomplete, non-standard, and open to misinterpretation. They should have just referenced the Monster Manual entry as does Conjure X.
The objects in question would have no attack or damage associated with them on their own. This is not like the boulders in the Earthquake spell, as these are attacks. Therefore, in this case, the attack or damage must be coming from the ongoing spell effect, a magical source.