Hardware audio mixing
This is pretty much how modern sound cards work: 

Following the description, Paula simply gets audio as PCM samples, like those stored in an (uncompressed) .wav file.

So, there's nothing inherently different to "modern" audio hardware.


  How these 4 input channels can be translated into modern DSP world?


Whilst not necessary, many higher-end modern sound cards do support some form of hardware mixing.


  Is there any analog mixer chips with multichannel support and how it would work?


An analog audio mixer is really just a (weighted) signal adder and can be built with any opamp and a low amount of external resistors.

You could control multiple programmable-gain amplifiers and use the adder to mix the result, but that'd be a waste of money; you'd just do the mixing (and that is really just addition) in digital domain wherever possible.