What are the origins of the phrase "operate on register[s]"?

I would venture to say that the phrase has not yet gained widespread usage at all in the sense you describe. For instance, a search for the specific phrase you cited turns up only a handful of instances.

Setting that aside, I think we can still look at the changes in usage over time.

A simple Google search for "on a number of registers" turns up many examples, but a lot of them are specifically in the sense of operating on computer registers. Skimming the list turned up only a few instances of the sort of usage you're asking about.

An NGram search shows that the phrase "number of registers" has, in general, followed the prevalence of computer registers - increasing in the '50s and then decreasing in the '90s (as low-level assembly language programming fell out of favor).

Skimming NGram results pre-1950 shows only a few instances. Most of them are around voting registers of some sort. Again, none of the sort of examples you describe.

So based on all that, I would conclude that: Yes, the use of "registers" in the sense of your example, while not exactly common, is something that evolved concurrent to the advent of computer registers.

Edit to add: When I say that they evolved concurrently, that is not to imply that this usage has anything to do with computer registers. As MetaEd points out in the comment above, the OP's usage comes from music. The OP's question was whether it "gained widespread use" after the advent of computers or not, and it seems that it did gain in use after the advent of computers.

I doubt this has anything to do with computer registers specifically. Computer programmers inherited the term from electrical engineers (example), who themselves borrowed it from mechanical registers (such as cash registers), which derived their name from paper registers, in the sense of a book of records. All of these are just different technologies to represent the same concept: A place where a piece of information can be stored and operated on.

Even today, more people are familiar with paper registers or cash registers than computer registers, and even fewer would have understood the term as computer-related in the 1950s.

I don't think the musical metaphor is the correct explanation here. The notion of a register of memories or beliefs is still fundamentally in the domain of information processing, and metaphor is not required to explain the use (nor is it clear to me why the metaphor would apply). Also, musicians do not say "singing on the high register," but rather "singing in the high register," so if it were a musical metaphor, the given usage would not be correct.

Drawing from one of the linked examples (the Henry Giroux article), he writes:

Cinematic violence operates on many registers...

but later in the article he also writes:

Motion pictures depicting "realistic" portrayals of black ghetto life add fuel to the fire by becoming a register in the popular mind for legitimating race and violence as mutually informing categories.

It's clear to me that, for this author at least, this use of "register" comes from its information-related definitions rather than its musical ones.