Hello CC Community,
Here are my mathematics related stats:
GPA: 3.65, ACT: 35 (36 in Math), SAT II: Math 800, AP Calc BC: 5
I have heard from teachers and counselors about the rigor of getting into college as an engineering major (some counselors have even said that declaring yourself as an engineering major reduces your chances of getting into that college).
Hypothetically, say I were to apply to Berkeley as an EE major. Will my chances of getting admitted be lower or about the same if I were to apply as an applied math major?
thanks. 
It tends to be for those schools that have separate admissions for different schools. It may occur to you that people could say they were majoring in the less competitive major and then switch when they get there. Schools have that covered. It is a bad plan because you will probably be stuck with the original major. Schools make it very difficult to switch schools when it involves a less competitive to a more competitive school/major.
For Berkeley specifically, EECS is one of the most selective majors, due to its popularity. Applied math is in the College of Letters and Science, which does not admit by major.
Once enrolled, it is very difficult to switch into EECS later, because it is filled to capacity, and there is very little space opened up through attrition. In contrast, applied math is not a restricted major and just requires C grades in the prerequisites (though the department recommends that only students who earn B- or higher grades in the prerequisites declare the applied math or pure math major).
Some other schools do it differently. For example, UIUC and Washington allow applying to majors, but may admit as undeclared or to a second choice major students who are rejected from the first choice major but admissible to the school or second choice major. But then entering the first choice major later involves a competitive admission process based on college GPA and possibly essays. Schools like Texas A&M, Purdue, Minnesota, and Virginia Tech admit engineering applicants to a first year pre-engineering program; students must then compete by GPA and sometimes essays to get into their desired majors.
Clarification to the above: The only majors at Washington that have a “direct admission” pathway are business administration, music and other audition based majors and some of the engineering majors. Washington does not directly admit to “second choice majors.”