I'm a graphic designer/creative director, so things like this stand out to me. I just took a look at the cover again because I've been comparing a few "replica" T-shirts to buy (looking for the most "authentic" looking) and I noticed some interesting things about the "Prince and the Revolution" text.

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First, it wasn't a font, not that I believe existed in 1984. I believe this was hand-drawn original art that was later (hastily) made into a font.

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Look at the "Prince and the Revolution" line. The two "i"s, three "n"s and three "e"s are all different from each other. Some are majorly different, some are close but still unique. The two "o"s in "Revolution" are quite different than each other too.

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All of these little differences are not noticeable on the "Purple Rain" main title because no letter is repeated (save for the "r", which is not in the same title case so they're completely different anyway).

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It's a very unique lettering style, one that is instantly recognizable and fits this album perfectly.

Yeah it wasn't an existing typface. The fonts you can download around the web are a poor look-i-like and pretty naff tbh.