> Likewise if I take some GPL software, I can do whatever I want with it to run it

That's true, but unlike the DVD case it's so only because the GPL license explicitly grants you this right. If it didn't, you wouldn't be able to run it at all.

Based on what?

Based on copyright laws. Here in Poland you can even download a copyrighted movie or a song as long as you don't distribute it; you can't however use any software without a license that lets you do it. The GPL licenses explicitly do ("This License explicitly affirms your unlimited permission to run the unmodified Program.").

GPL is actually kind of a special case when it comes to licenses since it mostly cares about redistribution in order to implement its copyleft mechanism, but generally licenses are tools used to give users permission to use the software under specified conditions. Without a license (written or verbal), there's no permission to do anything whatsoever.

You may be confusing it with using a self-modified copy of code under GPL. Since GPL only cares about sharing the modified source with actual users, you won't have to share your modifications with anyone as long as you remain the only user of that modified work.

>there's no permission to do anything whatsoever

What happens if you do?

You're violating the copyright. In practice, usually nothing until the copyright holder decides to sue you for damages or it gets found during some inspection (but the latter is mostly applicable to companies).

Unsurprisingly I'm not familiar with the ins and outs of Polish copyright law, but in other countries the act of copying something into memory to use it (say copying a frame of video into a frame buffer on a DVD) is not a breach of copyright.

Neither it is in Poland (which you would already know if you read what I wrote more carefully).