Op Amp saturating below supply voltage?
Why is this happening?


Because physics. Op amps have an output stage made with transistors, and no practical transistor saturates to a perfect short. That's the simple answer, if you really want to know the physical reason, that needs to be a different question about solid state physics.


  What is the term for this offset that I can find in the datasheet?


It varies with manufacturer, usually it will be labelled output swing, and will be spec'd at some supply voltage and several loads, say \$ 1k\Omega \$ and \$ 10k\Omega \$.


  Are there op-amps that can go saturated all the way up to the supply voltage?


Because of physics, no op amp will get all the way to the rail. The rail to rail op amps mentioned in comments can get within about 50-100mV. If you absolutely must have 0-5V performance, you need to have a supply voltage slightly over 5V, and a negative bias generator.