rusty stains on tile in bathroom
There's a good cleaning solution for this called CLR (US) - it stands for Calcium-Lime-Rust. This will dissolve rust and hard water spots in showers, if you do indeed have rust.

You mentioned scrubbing doesn't help, if you haven't tried CLR I would do that next. Otherwise, you're probably looking at re-grouting.

It is possible to caulk these seams and that may work, but they're intended to be grouted. Caulk is not a good permanent solution here. Grout can be removed with a cutting/grinding tool, and it's relatively easy to apply. Color matching the existing grout may be difficult