Whilst I’m not sure I actually understand the question (as in Wheelz wondering whether the whole world experiences the same seasons at the same time, only with different temperatures) with the heat, stifling humidity, rain, and mozzies, yeah, I’d say it’s “Summer”…
An African once claimed about Sweden that we have two seasons: White Winter and Green Winter.
Sorry, I thought I was more clear. As Schnitte guessed, the question is simply about terminology. I was curious as to whether Austrailians (and other Southern Hemispherians) call January “Summer” or “Winter,” regardless of the weather. The factual answer came quickly and the subsequent discussion has been interesting.
Incidentally, I found that Cecil has weighed in with the observation that not everybody agrees when the seasons begin and end anyway.
The meteorologist are fairly sure when seasons begin and end (or at least how to define them). If the mean temperature over 24 hours have been below 0 C (32 F) for a period of seven (I think) days in a row it’s winter. If it rises to above 0 for a period of seven days it’s spring, above 10 C (50 F) it’s summer and then when it goes down to below 10 it’s autumn.
Mostly, Wheelz, being the pragmatic lot we are, January is called “January” - though it is of course a summer month. We tend to not refer to each month by its season, but rather its name.
Does that help?
I hope you’re just messing with me here. 
Of course I know January is called “January.” Let me rephrase again. What to you call the season in which January occurs? Answer: “Summer.” Got it, Greg Brady.
Yeah - I was messin’ with ya…it’s the heat.
Well, it’s 40 degrees here in North Carolina and down in Melbourne at the same time of year. Of course, here it’s Fahrenheit…! 
And Kuujjuaq (58°06′24″N 68°23′55″W) is actually south of Stockholm (59°21′N 18°04′E).
Yeah, I think much of it is also the fact that Europe largely has an oceanic climate, which leads to warmer winters and cooler summers, when compared to most of North America (except close to the coasts, of course).
It’s always fun to point out that I live North of Churchill, Manitoba, where polar bears roam the streets looking for garbage. 
In Newfoundland, the seasons are:
Almost Winter
Winter
Still Winter
Construction
Stockholm is actually the farthest north and the farthest east I’ve ever been; Arlanda airport is only twenty minutes of latitude south of being “north of 60,” as we say.
Yeah, except I’m posting from Newfoundland, an island in the North Atlantic, and it’s as cold as a polar bear’s bottom.
Not only does everybody not agree when the seasons begin and end; they don’t even agree that there are four seasons.
As others have pointed out, the in the tropical areas of Australia most people recognise only two seasons - wet and dry. In the temperate areas, the white population mostly recognises four, a habit imported from Europe, but the indigenous population have their own reckoning of seasons, based not only on weather but on what animals are doing, what plants can be eaten, and so on. Here in Perth six seasons are recognised.
Note that all or nearly all native Australian trees and shrubs are evergreen, so there is no season when the leaves fall, and no season of new growth.
I wondered about this, too, until saw Muriel’s Wedding (Muriel was played by Toni Collette, before she moved to Hollywood)—specifically, this exchange:
Sales clerk: When is the wedding?
Muriel: October.
Sales clerk (smiling) Ahhh, a spring wedding.
According to an Aboriginal calendar I have, right now in Sydney the season is Gadalung Marool, the time of the Kangaroo. This season starts when the wattle Weetjallen comes into flower and changes to Bana’marrai’yung when the lilly-pilly fruit starts to ripen.
There are 6 seasons in this calendar and they are all timed by the flowering of certain plants, or the behaviour of animals. Bit of a different perspective.
I can’t believe some of you didn’t know that’s how this worked.
Did you also not know that we can see stars that you don’t, and the moon is upside down?
In Panama, many trees lose their leaves during the dry season (verano/“summer”), especially in the drier parts of the country. Savannas and pastures turn brown and are burned.
Deciduous trees put on their leaves, the hillsides green up, and birds begin to sing and nest at the the start of the wet season (invierno/“winter”).
Holy cow, I’ve been here 8 years and I never noticed about the moon!
I need to go outside more.
Also, I have just outed myself as a moron, haven’t I? 
To the OP, as a Yank Expat, yes, it’s summer here when it’s winter there and yes that’s how we describe the seasons in Australia. Unless you live where it’s the Wet and the Dry, and that’s been covered.
I’m frankly surprised by the question; the meaning of the English word “summer” is “the warmest season”, not “that season in which January occurs”. But whatever.
I would say though that, the seasons being much less emphatic here than in much of the US, we do not refer to them nearly as much. From the US I hear greetings like “How’s your summer?”, and movie promos always seem to start with "This summer … " and so on. We never talk like that. Seasons are just not that big a deal here.