ASICs bake one algorithm into the chip. DSPs are programmable, like GPUs or CPUs. The thing that historically set them apart were MAC/FMA and zero overhead loops. Then there are all the nice to haves, like built in tables of FFT twiddle factors, helpers for 1D convolution, vector instructions, fixed point arithmetic, etc.

What makes a DSP different from a GPU is the algorithms typically do not scale nicely to large matrices and vectors. For example, recursive filters. They are also usually much cheaper and lower power, and the reason they lost popularity was because Arm MCUs got good enough and economy of scale kicked in.

I've written code for DSPs both in college and professionally. It's much like writing code for CPUs or MCUs (it's all C or C++ at the end of the day). But it's very different from writing compute shaders or designing an ASIC.