Saving things is creating an unauthorized copy, which is fundamentally the basis of copyright. Some territories have explicit allowances for personal copies but many don't.

There are a variety of reasons that you can have a copy of a copyrighted work it a way the copyright holder didn't intend, such as time shifting or media shifting.

And in many juristictions they are explicitly defined, and don't apply to copyrighted work generally, only when acquired under specific circumstances.

Format shifting is explicitly illegal in the UK for example, and timeshifting is restricted to works obtained from a broadcast source by the person performing the copy, and not to an on-demand one, in many territories.