I have heard natives say both
student with physics background
student with a physics background
so I assume both expressions are grammatical.
Is there any difference in meaning?
'Background' is a singular countable noun, so would usually need a word like 'a', 'the', 'my' or 'your'. So standard English definitely needs 'a student with a physics background'.
But there are some special contexts where articles and similar can be omitted, for example newspaper headlines and headings in articles etc (which are written sources). Imagine a university faculty committee decides to create a new part-time job for a laboratory assistant. They say 'We need a student with a physics background'. Then they write the advertisement, which starts 'New job available - laboratory assistant. Suit student with physics background'. You see the ad and you think 'I am a student with a physics background. I'll apply.'
Where have you heard native speakers saying 'student with physics background'?