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.
No, burning hands targets the creatures in its AOE
Burning hands is noted as having targets specifically in the DMG
The DMG specifically calls out the creatures affected by burning hands to be targets in an example:
For example, if a wizard directs burning hands (a 15-foot cone) at a
nearby group of orcs, you could use the table and say that two orcs
are targeted...
In the PHB there is even more support for the fact that creatures in an AOE are intended to be considered targets (more generally):
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell’s damage is rolled once for all creatures caught in the blast.
Flame strike is a cylindrical AOE spell that does not call the creatures caught in its affect "targets". However, the rules here clearly designate those same creatures as targets regardless.
Either way, the rules are clear: Creatures caught in burning hands' AOE are targets. And since the charmed creature would be targeting the charmer with a harmful spell effect, this is not allowed.
The Rules As Intended agree
Jeremy Crawford also seems to agree with this conclusion:
Dragon's breath has two sets of targets: the creature you give the breath weapon to and the creatures in the area of effect created by the spell.
Dragon's breath is a touch spell that does not specify in its effects that the creatures in the AOE are targets explicitly (just like burning hands). But according to Crawford, the intent is that those creatures are considered targets of the spell.
Also he gives more support here:
A typical area of effect has more than one target: the effect's point of origin and one or more creatures/objects.
The cheese factor
The point of the charmed condition is that the creature views the charmer as a friend and cannot/does not want to do any harm to them. As such, this whole argument seems like a really cheesy way to get out of the intent of this condition.
A charmed creature can’t attack the charmer or target the charmer with harmful abilities or magical effects.
Why is the character wanting to hurt the charmer in the first place? How does the charmed character justify deciding to purposely hurt their "friend"? How does the character know that this spell will work and not others?
As such, regardless of RAW above, I would not allow a player to purposefully pull off this idea at my table.
Best Answer
The creature does not automatically became loyal
The spell description says the affected creature "can't attack the charmer". This doesn't mean it can't attack the charmer's allies. This doesn't mean it became loyal to the charmer — it might still consider the charmer hostile. The spell only says it can't attack the charmer, and the charmer has advantage on social interaction checks, nothing more. Of course, a DM can decide otherwise.
Speaking of social interactions, the DM has the final word — it is the DM who sets the DC, so a player might easily succeed automatically or fail despite of the advantage.
More about the general idea on how spells work in 5e - What is the source of the "spells do only what they say they do" rules interpretation principle?