Do people living in the southern hemisphere still give their seasons the same titles as people in the northern hemisphere do?

Yes, they do.  They just have the opposite season to us.  For example, we are now coming to summer, but in Australia they are coming into winter.

I travelled to Australia when it was their summer. We left on 27th December and arrived to an exhausting heat.  All the Australians said it was their summertime and not to sunbathe over the lunchtime period, to drink plenty and always wear a hat.  Especially me, as I have extremely fair skin and can easily burn. 

I soon became accustomed to the heat and really forgot that it was our winter time (until we came home. It was certainly a shock to the system!)  That was until I walked past a shop window and saw pictures of Father Christmas on his sleigh, next to a snowman.  It was 30 degrees, so to see snow was a little strange. 

Not only there, my auntie lives in Argentina and she also refers to having her summertime over the Christmas period, and often sends me e-mails of wonderful BBQs when we are covered in snow.  I would be lying if I said I wasn't jealous.

It is due to the tilt of the earth, which causes the sun to be higher in the sky during our summer months.  In May, Jun and July, the Northern hemisphere is exposed to more direct sunlight as it faces the sun.  However, in November, December and January, the Southern hemisphere has its turn facing the sun.  I hope that makes sense.