Plumbing – What tool should I use for an octagonal nut on a plumbing trap

It appears from the video the sink bottom connection is assembled wrong. From top to bottom you should have sink, rubber gasket, metal flat washer, brass nut. This or the P-trap connection do not need to be super tight, firmly snug should be enough. If the connections leak when snug, the gasket or sealing ring is likely damaged, or possibly their mating surfaces are not clean and smooth.

More disturbing is you say drain/popup detached from the sink when you were fussing with the P-trap? This should not be possible. If it did happen, the drain/popup is broken and needs to be replaced. This is not something you could have done alone, it was ready to break anyway. The flange you see when standing over the sink is part of this assembly, the large nut underneath sort of clamps the sink between the rubber gasket and this flange. The drain/popup is normally installed by dropping it in from above. If it came out from the bottom it is broken.

Clean off mineral deposits with a brass or steel brush and a lime dissolver such as CLR or Limewaway. Towel dry as much as possible. Dry further with hair dryer in an attempt to evaporate as much moisture as possible; this, to leave room for penetrating oil to penetrate.

Aply penetrating oil such as PB Blaster or Liquid Wrench. Tap, tap, tap the the valve to set up vibrations to encourage the penetrant. A vibrator can be useful to assist, such as an electric tooth brush or electric engraving tool. Allow penetrant to sit for hours and tap some more after that.

Use a tight fitting wrench or possibly a pipe wrench to turn it anti-clockwise.