They don't have to be, say, from the same line of fixtures. But they should share some characteristics -- generally traditional or modern feel, sharp angles versus more smooth. There's a fair amount of freedom, but as a designer, I give it some real thought.

If possible I would select bathroom fixtures from the same collection. The accessories (TP holder, towel bars, etc) could be from a different collection, but complementary.

I used three different brands in my last bathroom remodel, but did keep the styling so similar that I don't think anyone would notice they weren't all the same brand unless they looked closely or were a builder or design professional. The shower fixtures are one brand, the sink is another and the tub something else (the filler came with the tub). I used chrome so the finish is identical on all of them as well. This was quite a money saver.

Check to see if the satin brass is the same on different brands before you go in that direction. Chrome and brushed nickel are very uniform across brands, I don't know if satin brass is. If you're going to have a glass shower enclosure you will also have to think about the clips on the sides that hold the glass. Those are usually the same finish as the shower fixtures.

They need to look like they belong together. The easiest way to do this is get it all from the same line, but you don't have to if you are skilled at making a mix and match that feels like it is within the same family.