I imagine it uses something like a SHARC DSP (which is what Strymons are based on, for example).
I was quite surprised recently when I opened my Meris Enzo (which I always thought was using one of those DSPs) and found out that it runs on an ARM M4

Regarding learning DSP, I am going to go with a shameless plug and suggest the Bela Real-Time Audio Programming course, this is more or less what we teach to our Master students.
You'll need a Bela (shameless plug #2) if you want to run the examples, but most of it can be applied to other platforms, you would just have to figure out how to set up your audio callback, hardware, etc
The Daisy might seem expensive, but if you compare them with a Teensy 4.1 and take into account the AKM codecs and SDRAM it doesn't seem too bad. I have one somewhere around my desk, but haven't gotten around building anything with it yet.
I can tell you one thing: manufacturing hardware at small scale is expensive