I thought they were testing her to see if she really was the mother by seeing if she could name someone the mother knew.

I think it was meant to be a red herring. We are meant to suspect that the 'mother' back living with them is the other girl in the photo. At least that's what I thought!

I guess it's something I've just never experienced or been privvy to outside of siblings. Queer.

It's not an austrian thing even I have done similar things, as did pretty much every kid back then.
It's just something that kids/parents sometimes do for fun, whether the kids have a similar hair color or just their height or something like that.
It's really baffling to me that some people (like you) are looking too much into this. I don't mean this in a bad way, only that this is something that's really, really common.
Dressing up just for fun. Just look at those 9gag posts where people find their doppelgangers and then they all dress alike to make a funny/weird/"omg what a coincidence" picture.
That's all there is to it, no matter how you try to rationalize it it just wouldn't make any damn sense for the mother to have a twin.
How come the kid(s) never mentioned their aunt, then? Why, during the whole course of the film, would they never mention "hey, but our mom has a twin, could it be her?"
And if there was indeed a twin whose existence was meant to be kept a secret, why would her picture be laying around like that, when even all the pictures with their father in them had been removed from the albums?
Doesn't make sense man. The picture was simply a red herring, just two friends having fun, probably abroad, where they bought/chose to wear a similar outfits, because that's what BFFs do girls just wanna have fun.
Like, think about weddings for example, where the maids of honor all usually wear the same dress and everything It's just for fun, for a special occasion, it's something that's really beyond common. People just do sh_t like that all the time.
Fine, fine, I'll leave! But first I'm going to bother these peanuts! Hmm? Yes? Hmm? HMM?

This is the only bit of the film that I can't rationalise in my brain. They looked far too similar to be just friends, and I've never known friends to dress exactly alike in that manner - is it an Austrian thing perhaps?

No significance whatsoever. Just two girl friends who dressed the same for fun on occasion, something that's very common, especially with girls.
Fine, fine, I'll leave! But first I'm going to bother these peanuts! Hmm? Yes? Hmm? HMM?