See photo:
Can’t tell if the undershirt suddenly gets wide and flat or what. (What do they call that style of undershirt in the business? I only know them as “wife beaters.”)
I’ve seen them called “A-shirts.”
It just looks like his shirt is starched to me. I wouldn’t guess body armor. Also, I can’t see his undershirt, but if I want to be less vulgar when referring to wife beaters, I just call them “tank tops”.
“A-shirts” are what they’re called on the label and in the biz, but then you’ll want me to start using the word “pant” or “tee” and it won’t stop until I"m buying $50 haircuts and talking like a fashionista. So I’ll just stick with “tank tops” and “wife beaters”, thanks. 
AKA Italian Tuxedos.
I once worked at a women’s clothing store (chain). When we got a big shipment of these (tons of different colors), the floor employees called them wifebeaters at the store meeting. The store manager was horrified and said, “you can’t CALL them that!” So we were like, “well, what the heck do we call them, then?”* It took her a good 30 seconds to think of another term for them (a-shirts, like you said).
Every single customer who asked about them said, “where the wifebeaters at?” 
- just “tank tops” was out because we had OTHER tank tops in a different cut and fabric that were called that.
I call them A-Shirts.
I think Obama’s shirt just looks bulky/flat because of the way the light/shadow is hitting it and because, as DrCube says, it’s starched.
I have no way of knowing, of course. I just think, since his shirt is kind of thin, it would be more obvious if it were body armor. Also, it seems to dip pretty low on the chest and have pretty narrow straps, as opposed to something like this.
If you have a laptop you can move the screen back to change the color/brightness ratios and clearly see what looks like a normal tank top. If it was body armor then it’s exposing way too much real estate.
Yup, take a look at this edited version. The undershirt has the clear profile of a wifebeater and not body armour.
In the UK we call them “vests”.