Your university certainly should count it as a mental illness. What have they said about it? Are they able to offer you any university support with it?
I've had severe social anxiety for a long time (same as you with school and being described as shy/quiet/etc!), but both undergrad and postgrad uni have been very good with me. Undergraduate uni basically let me get away with not attending anything (since lectures were not 'officially' recorded on attendance) and let me have extenuating circumstances for a presentation that I was unable to do. My current (postgrad) uni have been excellent - although I am not 'officially' diagnosed with social anxiety (for some reason it has always been worked on 'unofficially' in therapy) they have recorded me as having severe social anxiety on their SARA forms (statement of reasonable adjustments) which means that they can apply mitigating circumstances if my social anxiety has an effect on the quality of my work, ability to meet deadlines etc.
Tutors have also been great - at my current uni my first lecturer (and now diss supervisor) had a chat with me within the first couple of weeks about how I'd cope with presentations (I was only given the SARA back in February but it was pretty obvious anyway!); he helped me to communicate my thoughts by suggesting that I e-mail him before sessions with my thoughts on the readings and he'd bring them into the sessions for me, then building from that to trying to say a couple of words out loud if possible (wasnt a great success, but there we go...!) So yes, I have been lucky, had some very supportive people at uni.