It is definitely a fat slimline. Can't tell what type of wood it is but it is a hand turned pen.
for official documents POTUS chooses the style of pen he uses but they have come from Cross for a long time. Trump uses a Century II while Obama both Bush's and Clinton used the Townsend (Obama later switched to the Century II as well).
Cross has been the official supplier since Clinton but previous President's like Reagan and Nixon used Cross pens as well.
All of them have their signature either on the cap or body and are ceremoniously given as gifts to certain people after their use. That's why various pens are used for the same signature.
CNN had a great article talking about this a while back.
Trump sticks with tradition for presidential pen choice - CNNPolitics.com
You can buy Townsends and Century II's from Cross's website as well. I think it would be pretty cool if POTUS would use pens made by various turners to sign documents! Talk about supporting small business!