I heard that in some universities, if a school likes you but the university doesn’t, the school can fight for your acceptance. So if you apply to two schools in a university, would having two schools like you increase chances?
No.
When each college/school/program within the university has its own admission team, what matters is if the specific college/school/program that you applied to likes you. Institutions that have that kind of admission system can have very different expectations for applicants for the various divisions, and it would be unlikely that any given student would be equally interested and qualified for multiple divisions.
When there is one overall admission team, yes sometimes indivisual departments or programs or athletic coaches might be able to push a bit for a specific student, but that student would need to be known to the department/program/coach well in advance. It isn’t likely that that person would find the student in the middle of the application process.
At schools which admit by division or major, you can only apply to one, or you can apply to more than one in rank order, so that your second choice is only possibly considered if you are not admitted to your first choice.
Admission by division or major is commonly done in the following situations:
A. Some divisions or majors are “full”, and limiting enrollment in them is done at the frosh (or transfer) admission level.
B. Some divisions or majors have application requirements that are not used for others (e.g. calculus readiness for the engineering division, or portfolio for visual art majors, or audition for performing art majors).
In any situation where some divisions or majors are more competitive than others at frosh admission, you should expect changing into them after enrolling in another division or major to be competitive or require high grades or GPA.