I am a physics-math oriented student who has done many engineering projects in my time during high school, and I love mechanical and electrical engineering equally. Which one is easier to get into?
If it matters, switching into the harder-to-get-into one after enrolling is likely to require another competitive admission process.
Any specific school? I doubt there is a global answer.
And many schools have students not declare majors until the end of freshman year or even 2nd semester sophomore year. You just apply to the college of engineering.
I want to say EE but I’m sure it varies from college to college.
You need to check each specific school. It does look like, at colleges where EE does not include computer science and engineering (CS or CSE), EE is likely to be the less popular and less selective one if it matters (but there is no guarantee on that for a specific college without specifically checking that college). If EE does include CS or CSE, its popularity may be increased by those primarily focusing on CS or CSE.
Where students enter undeclared or engineering undeclared, also consider whether there is a secondary admission process to specific majors, and whether ME or EE is more competitive: http://talk.qa.collegeconfidential.com/engineering-majors/2174743-first-year-engineering-programs-secondary-admission-to-major-criteria.html
But really, you should be considering which whether ME or EE is better suited for your academic and career interests.