Press dies and shell holders

Completely interchangeable

No. In most cases you can interchange both as most major manufacturers make their dies and shell holders to the same standard. The only time you can't is when you're loading something that's large or out of the ordinary like .50 BMG.

Dies are generally interchangeable.

There are few exceptions, all the examples in the original post are interchangeable

All the current bench tools I know of except the Dillon Square Deal B use the 7/8x14 dies introduced long ago by the Pacific Gun Sight Co. and picked up by all other makes.
All the current single stage presses I know of use the T slot interchangeable shell holder, also introduced by Pacific. Progressive shell plates are specific to make and model.
Excepting, as above, monster magnum calibers and the shotshell tooling for a single stage.

Reloading equipment is pretty durable so you might run into some old equipment with different sizes.
Frex you can get an adapter to use T slot shell holders on Herters presses.
I wish I could get shell holders for my W.H. English.

An old Star Progressive is still a desirable machine, but it has its own dies and I don't know if anybody is making them.
There are Lyman presses that take the 5/8" dies from the 310 tong tool.

js6seaj said:

A friend asked me, and I'm about 99% sure I know the answer but thought I would get a second opinion. Can you for example use a Lee die in an RCBS press, or an RCBS die in a Lee press, or a Lee die in a Hornady press for that matter. My answer is no but never tied it, and highly doubt these parts are interchangeable.

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I'm guessing you've never read a reloading manual.

You should give it a try. It's amazing what you can learn by doing so.