How does Processing language relate to the industry standard? Here are its docs: https://processing.org/reference.
Processing is described as a 'language built with the purpose of teaching non-programmers the fundamentals of computer programming in a visual context'.  

However, how does this language relate to the real-world computer graphics tools?  

For example:  


Is it similar to some low-level library maybe?   
Is the concept of a repeating draw() { background(255) } (in order to draw each new frame of an animation) used in real-world libraries? 
Are the same
methods/concepts used (pushMatrix, popMatrix, rotate,
translate)?
It does not relate to low level libraries. Yes, usually each frame is drawn separately, nothing can change if i dont change anything. However the draw loop is often handled by somebody else in for example game engines, etc. So in fact processing is often lower level stuff than what many people would do day to day.

Pushing matrix state is pretty common, it exists even in graphics languages from 1970's.