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In the following problem, why must ∠BAE ≅ ∠CED? Can't ∠BAE ≅ ∠BDE as well if you simply flip the triangle on top around?

For instance: enter image description here

I assume it's not gived that $AB || DE$: it's not mentioned anywhere, and in the picture they aren't parallel.

In that case, it's only a matter of notation. When we write, $triangle ABD sim triangle DEF$, the order of the vertices matters, that is, $angle A = angle D$, $angle B = angle E$, and $angle C = angle F$. Similarly, in your case $angle A = angle E$, since they're written in corresponding places.

It is $angle BAE=angle AED$ since you are flipping the triangle along left-right(i.e. along a vertical axis such that it is proportional in size to the other, i.e. the angles are same and the side lengths are in the same ratio). If you're confused, its good to remember that in general

$triangle ABCsim triangle DEFRightarrowfrac{AB}{DE}=frac{BC}{EF}=frac{AC}{DF}$ also, $angle CAB=angle FDE,angle ABC=angle DEF,angle BCA=angle EFD$

If we assume that $ACE$ is a straight line (not explicitly stated, but a reasonable inference), then $∠BAE$ is the same as $∠BAC$. We're given that $Delta ABC ≅ Delta EDC$. When two triangles are given as congruent, that means that corresponding elements are congruent. It might help to write the letters in different colors, but apparently SE doesn't support that, so I'll use different typefaces: $Delta$ A B $mathbb C$ $≅ Delta$ E D $mathbb C$.

From this, we can conclude $∠$B A $mathbb C$ $≅∠$ D E $mathbb C$. Note that the typefaces match up: on the left, it's italics, bold, blackboard. On the right, it's italics, bold, blackboard.

Is $∠BAE ≅ ∠BDE$? Well, it could be. But the question doesn't ask for what could be, it asks for what must be. Writing that with my typefaces, (and keeping in mind that $AE$ and $AC$ describe the same line, as do $BD$ and $CD$) it's $∠$B A $mathbb C ≅ ∠mathbb C$ D E. The typefaces don't match up: the left is italics, bold, blackboard, but the right is blackboard, italics, bold. So while that doesn't prove that the two angles are different, it does show that their congruence isn't required.