Aeonium arboreum, possibly the variety 'Zwartkop' or 'atropurpureum' so no, it does not need a shallow pot, these can grow up to a metre or more. You won't be sure its Zwartkop until it's had some sun exposure - the leaves lose their black coloration without it, although in very hot and sunny places, they actually prefer a relatively shady spot. I wouldn't cut anything off the stem, unless its starting to show signs of rot from having been in water - there's a little root there already, although the stem is just a little thicker than one you'd usually choose to propagate from.
Take it out of the water and leave it on its side for a few days somewhere warm and dry until the wound at the bottom has calloused over and looks dry. Then pot up into a mix of sharp grit mixed with (preferably) soil based potting soil, making sure at least half the stem is above soil level. If you have access to perlite, after gently firming the cutting into the soil, spread a little perlite over the top to help keep the stem dry (or you can use a little sharp grit) then water. Instructions here https://www.gardenersworld.com/how-to/grow-plants/how-to-take-aeonium-cuttings/