Can two universities have the same name?

There are a number of Belgian universities which were one institution formerly, but have been split in two independent institutions during the language conflicts between the French and Dutch speaking citizens in the 1960ies. At the moment they in principle still carry the same name, but in the different languages. Examples are the Free Universities of Brussels (Vrije Universiteit Brussel and Université Libre de Bruxelles) or the Catholic Universities of Leuven (Katholieke Universiteit Leuven and Université catholique de Louvain).

Yes, for example there is a Heidelberg university in Heidelberg, Germany, and one in Tiffin, Ohio, USA.

Note that non-English names are sometimes rendered inconsistently in English, e.g. Ruprecht-Karls-Universität Heidelberg is sometimes called "University of Heidelberg", sometimes "Heidelberg University" (even on their own website).

Yes, two universities can have the same name.

Many universities are named after the place they're located in.

Many places in North-America, Australia, and New Zealand are named after places in Europe, in particular after places in Great Britain and Ireland.

Although I haven't done a complete survey, the chances would seem pretty high that this leads to a collision somewhere.

As ChristianW, OBu, and user2768 have all correctly pointed out, you should design your database such that a key is designed to be unique, not the university name.