See the excellent answer by Micheal Karcher, which is spot-on.
I'll add something about modelization, which is a concept that probably you still have to grasp fully, at least judging from what you say in your question.
Keep in mind that the formulas and graphs you use to describe, understand and predict the behavior of a component (and of any physical systems) are just mathematical models which have been validated experimentally to provide some given degree of accuracy.
The Shockley equation of a PN diode is a model, as well as the ideal diode model. And there are more complicated models too (for example those used in SPICE programs to model a real diode). It's just a matter of how accurately you want to describe the behavior of the diode.
As said by Micheal in his answer, the ideal diode model just describes the most basic behavior of a diode when used in applications where it is switched ON and OFF and where its voltage drop, or its dependence on current (and other factors such as temperature), are not worth considering.