Bing dictionary calls a transcript (definition 2):
student's academic history: an official document showing the educational work of a student in a school or college
If you focus on the transcript as a document, like a passport or a driver's license, then the student holds the document. But unlike a document, transcripts can have many official versions. In contrast, the student may, under normal circumstances, only have one official driver's license or one official EU passport.
The student could own or possess a transcript, just as she could own or possess a 1969 VW Beetle. (It's possible to think that a prospective grad school might also own the transcript by virtue of the student ordering a copy and sending it to the prospective grad school.)
If you focus on the history aspect, then ownership or possession sounds suspect.
One could just as easily say that the university owns the historical aspect of the transcript.
The answer to your question depends on what you're trying to express. Perhaps the simplest way to express the notion of possession would be:
The student has a transcript.
If you're actually referring to the degree:
The student earned a B.S. in Applied Physics.