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.