In Oblivion, 10% of your endurance level was added to your health at each level up, so I always tried to get my endurance up as fast as possible to maximize my health.

Are there any items, or other ways to give yourself the ability to have more than 10 points at each level up, such as something you can consume, or equip for example?

Nope!

Not on a console is the correct answer.

The only thing I've come across is something that allows you to add/subtract to your health/magicka/stamina at will. Is there another command that changes the set amount gained per level? owo

I do not know of a mod that can do this at every level up, I usually to avoid using mods that do this. Playing around with the mod Levelers Tower has been a bad enough of a distraction from actually playing Skyrim

The Skyrim Uncapper can be used to change the amount of health/nagicka/stamina that you get each level.

I am assuming this "uncapper" is only on PC?

Yes.

Simple answer:

PC - Yes
Consoles - No

Fixed!

Fixed
Okay, I'm outta here

The 10-point increase is preset, thus cannot be modified in-game, but if you're on a PC it can be changed in the CK. I recommend not doing large changes to the amount, though, as even a 5-point increase is a couple of hundred extra health by level 41, which is a bigger buffer than one might think.