Looking at the definition of the Euler number leaps to mind! The number "e", the base of the natural logarithms, is sometimes referred to as "Euler's number" and, in addition, the limit, as n goes to infinity, of
is also referred to as "Euler's number".
But here, you mean what is more often called the "Euler characteristic". For a plane figure that is "the number of faces minus the number of edges plus the number of vertices": F- E+ V.
The digit "0" is topologically equivalent to a rectangle which has 1 face, 4 edges and 4 vertices: F- E+ V= 1- 4+ 4= 1.
The digit "1" has 0 faces, 1 edge and 2 vertices: F- E+ V= 0-1+ 2= 1.
The digit "8" is topologically equivalent to two rectangles sharing a common edge. It has 2 faces , 7 edges, and 6 vertices: F- E+ V= 2- 7+ 6= 1.
The letter "A" has one face, 5 edges, and 5 vertices: F- E+ V= 1- 5+ 5= 1.
The letter "B" is topologically equivalent to "8" and has the same Euler characteristic: 1.
The letter "C" is topologically equivalent to "1" and has the same Euler characteristic: 1.
The letter "D" is topologically equivalent to "0" and has the same Euler characteristic: 1.
The letter "E" has 0 faces, 5 edges, and 6 vertices: F- E+ V= 0- 5+ 6= 1.
In fact, the Euler characteristic of
any such polygonal figures is "1"!
That fact that this does not match Captain Black's answers makes me nervous but I am going to stand by my answer. Captain Black's reference to "bodies" and "holes" makes me think he is interpreting these figures as solids in 3 dimensions, rather than figures in the plane.