What’s powering those cards & how powerful are they?
I did a bit of research. Here is a page that sells these things. From this, and because the usual 8 bit cheap cpu (or even a 16 bit one) doesn’t have the signal processing capability needed for these, I think they are custom chips. Very cheap, and the gigantic volumes cover the design costs. Since they are so small, and don’t need advanced technology, you get lots per wafer, which makes them a good deal. There is no doubt a cheap cpu in there also.
Most cpus sold aren’t the ones in PCs, but rather the ones in microwaves, etc. Here is the Wiki article on them.
I hope that more or less answers your question.
Voyager is correct. They are usually custom chips. The computing power is generally very small. The chips are mostly a memory with an A/D converter and a small circuit to sequence through the memory locations. Others do essentially the same thing, but use a tone generator instead of an A/D converter. It takes a lot less space to store notes (kinda like sheet music) instead of digitizing the sounds.
Here’s a couple to look at:
http://www.winbond-usa.com/en/content/view/36/1518/
It’s probably just as well that the CPUs in musical greeting cards aren’t too powerful, lest they suddenly gain sentience, adopt Marxism, and try to destroy humanity… 
I know, it’s GQ, but come on, the OP is just begging for a Futurama reference…
On this theme you might enjoy “Naught for Hire” by John Stith. It was in Analog, I can find it if anyone cares - I don’t know if it has ever been anthologized. This is a detective story in a world where there is a bit too much AI in appliance.
The hero is in an elevator, and the doors won’t open. “Sorry Dave, I can’t do that” says the elevator.
I’ve read about seven novels by him thanks to that story.
Right, they’re called ASICs.
More likely their modern equivalent, Systems on a chip or SoCs. ASICs don’t have a processor, SoCs usually do. SoCs consist of a number of blocks, called cores (nothing to do with core memory) which you can buy from various places. The biggest vendor of CPU cores is ARM, and they also often include memories, DSPs, and core to handle various flavors of I/O. You can even get open source cores at opencores.org.