when applying to my dream school, should I mention that it’s my dream school on the application?
No.
Have a dream life plan, not a dream school
NO. There are way better ways to discuss WHY this school is a top consideration that dubbing it a “dream school”.
That is how I feel also.
If you even have a dream university, you should have a very good reason why it is your dream university. You should understand what is different about that school and what factors makes that school a good fit for you. If you know this, then you can talk about these factors (whatever they are) without using the phrase “dream school”.
I did have a “dream school” when applying to graduate programs. I did not get in. I went to a different university which was in fact higher ranked for my major and turned out to be a better fit for me.