This is ambiguous, and you can claim one interpretation, while the university will claim the other interpretation. Ideally, Customer should verify before buying ! CAVEAT EMPTOR !
In your case, it says "university gym summer membership", which implies the limited nature of the Subscription. Without the word "Summer", the university is wrong & you should get your 2 months. With the word "Summer", the university is right & Subscription is limited by some terminal Date.
The web-site might have listed the T&C, which might explain the situation. If not, the university can still get off by Interpretation, & they can update the wording to avoid such things. You should talk to them Directly.
Valid Scenarios where interpretation can go either way:
A week-long celebratory event : You can buy Daily tickets @ $1 or Weekly tickets @ $5 (valid till Day7), even though there may be only 3 Days remaining. You cannot buy the Weekly ticket on Day6 and claim that the event got over in 2 Days !
Discounted Yearly Subscription to magazines : I have come across cases where I got 50% Discount, but only if I subscribed within two months (Eg valid till July 31); Else the Discount expires. This does not mean that the Subscription is for 2 months; only that the Discount Availability is for 2 months.
A 6-month Course (July to Dec) : You can enroll in the second month or in the middle, but you still have to Pay the whole amount & you can not claim that classes should continue for 6 months.
Yearly Mobile Connection : I can renew at old rates (valid till June 30) before the rates increase. I expect that I still get the 1-Year Connection, not the 1-Month Connection.
Putting all together, it looks unlikely that the university is going to return your money or give you more time at the gym.