Kind of. The "tax" is a compensation for those cases where you are allowed to copy without asking the copyright holder. It's called "Privatkopie" and means you are allowed to give copies to your friends and make copies for your own private use.
And downloading music, movies,... is not illegal in Austria. Upload (distribution) is what the rightsholder has the exclusive right for.
Games (Software) is a different case. The tax has nothing to do with that.