Feat for combining animal companion progression for Barbarian and Druid?

Is there a feat that allows the combining of the progression of the animal companion feature of the Barbarian (Mad Dog Barbarian) and Druid (Arctic Druid) classes for one specific animal companion?

technically no - Boon Companion would allow 4 levels of one of your classes to count towards your animal companion from another class so that might accomplish mechanically what you are seeking - but would technically by RAW require you either be another archetype of Barbarian or to have a domain not an Animal Companion as a Druid (which is probably would I would suggest - do a build like Barbarian X Druid 4 gaining your animal companion from your Barbarian levels (as a Mad Dog Barbarian) and then take a domain in place of an Animal companion for your Druid levels - and take the Boon Companion feat at an appropriate level (perhaps level 5? if you do 4 levels of Barbarian then 1 Druid - take Boon Companion and your companion will keep leveling up with you - or you could take it at level 3 and then alternate barbarian and druid levels (though honestly since Barbarians can be really strong at low levels you might find the HP and BAB from low levels very helpful)

For non-PFS play as a DM I would almost certainly work with a player on a concept like this - I'd either rule they could accomplish this via a variation of the Boon Companion feat - or I might just handwave the rules and let their Animal Companion granting classes stack for the purpose of determining their Animal Companion and either impose a small balancing factor/restriction or I might just say that it results in a far cooler character (and an easier table for everyone to run - i"d rather one PC, one Animal Companion as a DM vs One PC w/two or more Animal Companions)

Dammit. Boon Companion seems so troublesome as a feat (I wish they had made it simple and easy and just allowed the combination of two different specific classes's animal companions). Oh well, do you know if there is a winter-based domain that a Druid can take? If it's a subdomain, can a Druid (Arctic Druid) take something like that or are subdomains only available to clerics?

Ok, I am not sure what you are wanting. Animal companions always stack.

The character's druid level. The druid's class levels stack with levels of any other classes that are entitled to an animal companion for the purpose of determining the companion's statistics.

I don't see anything in the arctic druid that modifies the AC so with Mad dog and Druid the animal companion lvl will be equal to the combined level of the two. you don't even need boon companion.

What I'm wanting is one single animal companion with a effective druid level that combines both my Barbarian's class and my Druid's class (instead of two separate animal companions with each it's own effective druid level).

Oddly the Mad Dog Barbarian doesn't have the language about Animal Companion levels stacking - but since the Druid does then in this case Paizo's RAW already does what a reasonable DM would do - let your levels stack and combine so you have one animal companion.

(in which case disregard my advice)

The flavor of the Mad Dog implies that it was intended that the Animal Companion you get is a Dog (or perhaps a wolf) re the abilities you get, Pack tactics etc. But the actual archetype mechanically seems to allow you to have any Animal Companion option open to Druids in your campaign. As a DM I might actually restrict the Mad Dog Barbarian's options for animal companions more than I would a pure Druid - but for PFS etc it would appear you could take anything you wanted.

So no need for Boon Companion (though I think technically if you wanted a domain open to Arctic Druids it would be an option open to you)

That's what he's saying. when you have multiple AC granting classes, the levels stack for all of them per the druids AC entry in the CRB.

Core Rule Book wrote:
The character's druid level. The druid's class levels stack with levels of any other classes that are entitled to an animal companion for the purpose of determining the companion's statistics.

Mad Dog grants an AC ... so any druid levels will stack with the mad dog AC feature... no feat needed.

What I'm wanting is one single animal companion with a effective druid level that combines both my Barbarian's class and my Druid's class (instead of two separate animal companions with each it's own effective druid level).

That is in fact the rules for how it would be. In fact you can't even choose to do it the other way.

Sub-Domains are open to anybody, albeit they are limited per Deity.
Druids don't seem to be subject to any limitation on Sub-Domains though, beyond the full Domains they derive from.