So the rule for insanity states...

"insane characters have the following statistics: 0-cost, 0-skill, no
faction, no icons, no traits, and no text. An insane character can-
not commit to a story and is never considered to be committed
to a story."

It does not say anything about "name". This being the case, If an Alaskan Sled Dog is made insane does it reduce the combat of all the other Alaskan sled dogs in play or is the name on the card the only attribute which is retained during insanity. If this is the case then insanity would not effect the #'s of the other dogs in play.

The larger question:

Are cards that go insane consider to be an neutral insane cards with no qualities what-so-ever, or do they retain only their identity (consequently their status as "unique" as well)?

as you said

"insane characters have the following statistics: 0-cost, 0-skill, no
faction, no icons, no traits, and no text. An insane character can-
not commit to a story and is never considered to be committed
to a story."

So they have a title and still count as unique. The text for the Dogs is "...for each card titled Alaskan Sledge Dog in play". So you gain icon and skill for all dogs in play (insane or not).

Alaskan Sledge Dog
------------------
Type : Character
Cost : 1
Skill : 1
Icons :
Subtype : Creature.
Game Text : Immune to Polar events. You may play up to 8 copies of Alaskan Sledge Dog in your deck. Alaskan Sledge Dog gains a C icon and +1 skill for each card titled Alaskan Sledge Dog in play.
Flavor text :
Illustrator : Leonardo Borazio
Collector's Info : AMM F16

And likewise, you can't bring out a second unique character when your first is merely insane...

On the other hand, if you have a Heroic character who's insane you could technically play (or take control of) a Villainous character without having to discard either (or the other way around) - so long as they remain in the loony bin...

Hmm, I was not aware of this. I thought a character being insane removed their title as well.

****, I'm gonna have trouble with my friend's Sledge Dog deck.

lol, on a slightly different note we have sledge dogs and they are indeed insane !!