If you reference specific things that you like about the school, you aren't being generic.
In the flowery version of 'I like Y school because I am passionate about xyz for reasons abc and this fits perfectly with their Z program', xyz and abc may not change when Y does, but as long as Z changes and references something that is specific to Y school (even if other schools offer analogous programs), you're not really being generic.
If you say "I like Y school because I am passionate about xyz and want to be a doctor", you aren't referencing anything specific to that school and THAT is generic.
Basically, if your essay could simply be copied over, word for word, to another school without any edits other than school name, that's when you should probably dig a bit deeper.
I've been tying everything back in to my already-established interests. I'm interested in physician advocacy and have the resume to back it up, so I'm applying to a lot of schools with advocacy/health policy program tracks. Some schools are located near hospitals with strengths in a particular patient population that I've really enjoy working with and would like to work with in the future- I mention that, too.
Try to relate your background/experiences to the opportunities offered by the school. For example, if your volunteer stuff relates to underserved people, and the school offers a free clinic (most of them do, FYI), then talk about what you've learned from volunteering that would allow you to contribute to the free clinic.
I tried the whole "find specific faculty and praise them to the high heavens" route before, and I don't think it was all that successful, but that's just me.