I think you misunderstand exactly what an odds ratio is. To see why it's probably not the best way to answer your question requires a tangent into exactly what odds and odds ratios are. The simple answer is that an odds ratio is a ratio of ratios which isn't really as complex as most make it sound.
To begin, you must first understand what odds are. It's simply the odds of success to the odds of failure. This is similar to but distinct from probability. So an odds of 1 means a 50-50 chance of success. An odds of 2 would mean you expect 2 successful outcomes for every unsuccessful one. An odds of .5 would mean half a success for every failure or, put differently, 1 success for every 2 failures.
With this in mind, an odds ratio tells you how much you can expect the odds of your outcome (your dependent variable) to change for a 1 unit increase in your independent variable. So, an odds ratio of 2 for your independent variable would mean the odds of success double for every one unit increase of your independent variable. From this, you should now wonder what the original or base odds were because doubling a small number is not the same as doubling a large number. Unfortunately, most never consider the baseline odds when interpreting odds ratios which is most unfortunate.
Now that you hopefully have an understanding of odds and odds ratios you should be able to see that they are most useful for understanding dichotomous outcomes - things that can be quantified as success vs. failure. This could be something like pass vs. fail, dead vs. alive, incarcerated vs. not incarcerated.
From your description it sounds like the outcome you're interested in is categorical but not dichotomous which makes it somewhat less intuitive for study using odds ratios.
It is still possible though. If there's an ordering of the categorical variable then you can generate odds ratios which predict the odds of being in a higher category vs. a lower category. If the categories are not rank-able then you can pick a base or reference category and compare the odds of being in any other category vs. that one. However, these approaches aren't very intuitive and there's probably an easier way to understand the relationship between university and your outcome, perhaps a contingency table.
In any case it would help to know more about your data and research question.