The open strings, on the keyboard
Open strings G, C, E and A.
The open strings are G, C, E and A, from low to high. The gaps between them decide whether a shape keeps its form when you move it across the neck.
Where the next string begins
3 string pairs. The odd one is C to E, which is fret 4.
Each ringed note is played on one string and matches the open note of the next string up. On every pair but one, that note is at fret 5. The exception is C to E, which is fret 4. The strings are tuned in two fourths and a third, and that one third is why a shape learned on the low strings comes out wrong on the top two.
A ukulele has four strings, and the gaps between them are not all the same. From G to C is five semitones, an interval called a perfect fourth. From C to E is four semitones, a major third. From E to A is five again, another fourth. That makes two fourths and a third, with the third in the middle.
That one uneven gap is why a shape you learn on the bottom two strings comes out wrong when you move it to the top two. The E string itself is not the cause. The tuning is uneven in exactly one place, and every shape in this course either crosses that gap or stays on one side of it.
A guitar has the same uneven gap, between its G and B strings, which is why guitarists talk about the B string. On a ukulele the gap falls right in the middle of your hand.