It looks like some want you to apply to a specific school whereas some colleges ask to list top 2 majors. In the latter case, if the majors are in different schools, is it possible to get a sense of whether one of them is a better chance? Is such data even available anywhere? For example, engineering vs a physical science major. Thanks
It depends on the school but there are some that do publish admit rates by at least college.
FWIW, my daughter thought major was more important than the school so she didn’t apply to a second major anywhere because typically those schools make it very hard to transfer into the impacted majors.
As @momofboiler1 mentioned, some schools do report admittance rates and average GPA/test scores by major.
At most schools, engineering would be a lower admit rate than physical sciences.
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Sometimes, you can look up the process to change into each major at each college. If the major is filled to capacity, it may require a high GPA or competitive admission to get into it as a current student.
If that is the case, then direct admission to the major is likely to be more selective for frosh applicants as well.
Between engineering and physics or chemistry, if there is a difference in selectivity, engineering is more likely to be the more selective major.
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I wish more kids I encountered online thought like this. If you are truly indifferent that is fine. But if you really want to try out a program with special admissions, I think you need to make it clear to colleges that is what they need to offer you.
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College admissions counselors can usually tell you which of their programs are the most competitive. Don’t be afraid to reach out with a direct question.
My kids both applied to both first and second choice majors not because they were indifferent, but because they had already observed themselves changing their minds a lot, and were honestly not sure if they would still prefer the first choice major by the time admissions decisions rolled in. So the idea was that it might be good to have a mix of majors to choose from in the end. 
So I also think more kids should understand that most people change majors in college, sometimes radically, if they can.
But I think that points to the sort of programs where either you don’t declare a major until later, or at least it is easy to change majors. Because even your best two ideas in HS may still not include whatever ends up being your graduating major in college.
My comment above was intended instead for cases where there is first-year admissions to the relevant programs, and it would be difficult to do an internal transfer later.
If you happen to be applying to Virginia Tech, you can do this sort of sorting on their websit (here: https://udc.vt.edu/irdata/data/students/admission/index#college) - you can look at admissions data by college, narrowed by major, narrowed by gender, narrowed by whether you are in state or out of state or international. You can even add in first gen or veterans status. It’s a very handy tool. I’m sure there are other schools that have this available, although none of the schools my kids applied to had it (at that time) so this is the only one I’m aware of.
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Sort of using a similar approach since multiple majors can land you in the same field/industry and also it sounds like Engineering admissions are harder across the board.
Engineering admissions tend to be relatively more selective than admission as a whole for the university. Also, I think that in the past year there seemed to be a trend for some high stat students, who in previous years would have been applying to CS majors, to instead apply to an engineering major such as mechanical or electrical engineering. (The demand for admission to CS majors has decreased significantly.) So although we do not yet have all the data for the past admission season, I have the feeling that engineering majors may have become more selective than before. We will see when the data is available.
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This seems to be more common at larger more selective universities that are not awash with money. I.e. many of the more popular and selective state flagships, where the number of students interested in and capable to completing engineering majors is relatively large to the point of filling up the engineering departments’ instructional capacities.
Some of the most selective private schools are awash in money, so their engineering departments have ample instructional capacity for all interested students who make it through the admission gate to the school. Less selective schools may not have enough interested and capable students to fill up the engineering departments’ instructional capacities.
I have the same impression about the different ways engineering admissions can work. Indeed, at some institutions, engineering majors are just on the list of ones you can simply declare later, they don’t even have separate first-year admissions. Even if there is some sort of first year process, at some institutions it may be easy to internally transfer into engineering. As you suggest, all this implies a lack of capacity constraints in engineering.