Is it normal to break out in spots after eating take away pizza?

That sounds a lot like a food allergy you must not know you have. It's very unlikely that it has anything to do with it being take-out... if you were to eat the exact same pizza in the exact same restaurant where you're getting your take-out, I'm betting you'd have the same reaction.

The basic ingredients of pizza are very simple and commonly found in other foods, so if pizza is the only thing that seems to trigger it, it seems likely that it's either a topping you get, or something you're eating (or drinking) along with the pizza. Some people have an allergic reaction to wheat, though I would think you'd observe that with other foods too.

If it is an allergy, you definitely don't want to treat it lightly, since it could create a severe reaction at some point, and can even be fatal - particularly if you don't even know what it is to avoid it.

If you can afford to, I would definitely go to see an allergist, and get tested to see what things you are allergic to. Whether or not you can do that, at the very least, I think you should start carrying around a benadryl tablet or two, just in case, and take one as soon as you observe those symptoms next time. It's fairly harmless if you don't need it, and could literally save your life if you do (it might make you drowsy for a little while, so be careful if you have to drive).

Some people can get spots from certain foods while others don't. My guess is you have a milk/dairy allergy. I know with me if I eat dairy within 1-3 days I get red inflamed spots which sometimes itch on my face and chest. It looks like acne but I think its more like hives.
Humans aren't supposed to eat dairy anyway so the answer is yes its normal if you are sensitive to it. Your friend might eat it and be fine but were all different.

It might be a food allergy. Stop eating take away pizza or buying it from a store and make it from scratch.

It might be cause it's super oily.

like a hyena?

Well stop eating it lol.
Seriously though..good advice above.