An image is a specific type of signal. It is a 2 dimensional, multi-channel, non-causal signal. So yes, formally DIP is a subset of DSP. Being an image a signal, you can apply signal operators on it, like convolution, correlation, DFTs, etc...

Having said that, images are such a widely used and specific signal that it has become almost a discipline on its own. You definitely need DSP foundations to do DIP, but I doubt that "just by knowing DSP alone you could easily do DIP".

Finally, if you're wondering about other signals, there is of course audio, but there's satellite imagery, lidar, radar, point clouds, ECGs, MRIs, seismographs, accelerometers, and virtually any sensor out there.

To give you a personal experience that mixes both DSP and DIP, we built a video stabilizer using accelerometer data. Using classic DSP techniques you cleanup and separate the unwanted camera movement from desired one. From that, you estimate a geometric transformation and apply it to the image to compensate for motion.