What is the technical term for the product of "number of pixels" and "number of channel per pixel"?
number of channel per pixel


This is typically referred to as colour depth or bit depth and expressed as bits or 'bits per pixel' (bpp)


  number of pixels


There is no special terminology for this, it's just "Pixel count" or similar. It's unusual to need to refer to the total pixel count of an image without any other context, the dimensions are usually used instead.


  If I calculate 3mn, what is the name of quantity for this value?


There is no special word for this quantity, but the units would typically be bits or bytes, for example

1024 x 768 x 32bpp

= 3,145,728 bytes (3Mb)