If I subscribe, is there any chance my university will find out I am using this site?
Your question raises a few issues I think should be addressed. These concern privacy, academic integrity, and practicality.
First, enotes administrators do not provide a "list" to any university about who is using the services. Nor do any of the educators who answer questions and prepare materials for the site. If you decide to subscribe, when we answer your questions, we will not even know what university you are attending.
Second, when you make use of the help enotes provides you, it is meant to be in a way that is academically honest. For example, if you quote or paraphrase something you read on the site, you are expected to document your source of information, with both in-text citation and on a references or works cited page. This is true of any source you use, no matter what, and a failure to do so is plagiarism. I have provided an enotes link for you below, that explains how to avoid this problem. If you have a professor who says you are not to use a particular source for your research and writing, as some professors do, for example, barring students from using Wikipedia, then it is academically dishonest to use that site. I am not aware of any professors, though, who do not allow the use of enotes.
From a practical point of view, if you have a username that identifies you and provide information that lets everyone know what university you attend, which I have never seen happen, no website can ensure that there would not be someone who knows you and your school. So, as is true for most on-line activity, if you wish to remain anonymous to others, you have the power to do that.
It would be a shame if your concern stopped you from using such a valuable resource. You do have a great deal of privacy. You should be letting your professors know what your sources are. And you have the means of controlling what others know about you. Give it a try, and I am sure you won't be sorry.