There are items that have powers and properties that look for fey origin:

• Cold Iron Bracers grant a defense bonus against fey creature's attacks.
• Cold Iron Weapons' immobilization daily power damages fey creatures.
• Fey Slaughter Weapons deal extra critical damage against fey creatures.

I can't think of any feats that require "fey" but I could imagine such.

I can't think of any monsters that have powers (or class powers) that look for fey, but again I could imagine such.

RAW - No, Otto's Irresistible Dance is not a Charm spell

It does not say that the target will be "charmed", and the condition it imposes is significantly different than the description of the "Charmed" condition.

OID:

A dancing creature must use all its movement to dance without leaving its space and has disadvantage on Dexterity saving throws and attack rolls. While the target is affected by this spell, other creatures have advantage on attack rolls against it.

Charmed:

A charmed creature can't attack the charmer or target the charmer with harmful abilities or magical effects. The charmer has advantage on any ability check to interact socially with the creature.

A dancing creature can attack the caster (with disadvantage -- and that disadvantage applies to any attack, not just against the caster) and can cast AoE spells that include the caster and anyone else they choose in the target area with no penalty. They just can't move freely. A charmed character can move freely but can't attack or target the caster.

Furthermore, if it were a Charm spell, it would not be necessary to specify that "[c]reatures that can't be charmed are immune to this spell." That appears to be listed as a special exception.

RAI - No, Otto's Irresistible Dance does not impose the Charmed condition

Jeremy Crawford has tweeted saying:

Being charmed means being subjected to the charmed condition

Since OID does not impose the Charmed condition, the Fey Ancestry resistance to "being charmed" does not apply.

What makes sense?

Webster's dictionary says:

charm: to affect by or as if by magic : compel

Magically forcing someone to dance seems to fall completely within that definition.

I would give those with Fey Ancestry advantage on saves against all enchantment spells that compel the user to do something that is not their own free choice. This includes many spells that do not specifically say they are "charms" or that they impose the "Charmed" condition, such as command, compelled duel, and yes, Otto's irresistible dance. I believe those all fall within the common sense and dictionary definitions of "charmed" in this context, and there's nothing in the description of Fey Ancestry that says it only applies to things that grant the "Charmed" condition.