What and why this type of cast iron pipe behind my sink?

it's a loop vent

it's a loop vent that is approved usually for island sinks, it doesn't have to be that high, usually they are plumbed just behind the sink or behind the flood level rim

vent loop for a foot vent

It's either a major drain line or part of the vital Vent Pipe System that removes methane gas from your home.

easiest way to think about this loop vent is a lot like a bottle of laundry detergent. one end you have a valve you press to release the soap. The other is a screw cap. if the cap is tight? no air gets in and you get zero to very little flow out of the valve. crack open the cap and let air in? you get free flow out of the valve. Same purpose here.

If you have to ask all these “experts” on here, you should not be I. The constitution fields…

Maybe it's the discharge for the dishwasher

Obviously it’s not the first remodel.