The Student license does not expire. However, it is only valid while you are registered in an appropriate degree granting institution. For example if you went to work for 3 years and then went back for a Masters, then during those 3 years you would not be entitled to use the Student license, but you would be entitled to start using it again while you were working on the Masters.
There is no upgrade path to Professional license from Student license: there is only purchasing the Professional license. The license costs can add up fairly quickly, especially if you get MATLAB Compiler or MATLAB Coder.
Hobbyists can use a MATLAB Home license at a notable discount (but more expensive than Student.) Home license cannot, however, be used to earn incoming in any way, or for employment, only for learning and other personal use.
