Most of the universities in the US use abbreviations on their websites. Can it be considered a good practice to use abbreviation of the same university I am applying to?
The major issue with abbreviations is ambiguity. For example, OU could refer to Ohio University, University of Oklahoma, the Open University, and others. Other abbreviations like MIT are less ambiguous despite there being MITs in other countries. Similarly, longer abbreviations like UPenn are less ambiguous. Some universities, go by multiple names. For example University of Nottingham is often called Nottingham University, despite there being a Nottingham Trent University in the same city that goes by Trent.
When applying to a university, or for internal purposes, ambiguity is not an issue. If you are at, or applying to the Open University, in general everyone will assume OU means Open University and not Ohio University. In the rare cases where you are applying to the Open University from Ohio University, there could be ambiguity. Understanding the nuances of the name and abbreviation uses can be difficult. In general, you should probably steer away from abbreviations and just use the full name.
This is a good practice, but make sure to clear any potential confusion.
This will show you have done your research and you are familiar with the position you are applying to. However, your letter has to be understandable for someone who just begins recruiting in this field. I would advise you, to avoid any risk of misunderstanding, to use the full name the first time, and to put the abreviation you will use for the remainder of the letter between parenthesis.
Here is an example:
I am answering to the offer n°16112016 as a Car Builder (CB) for General Motors (GM). I believe GM is a great company .... and I will succeed as a CB because ... and so on.