The standard method used in EIGS is only efficient for matrices that are quite large. When eigs detects that the input matrix is so small that it will be more accurate and faster to just call EIG instead and extract the requested values from that, it does this instead and most options that affect the standard method's behavior are ignored as that method isn't used.
When a function handle is passed, this means that the function handle is just called on each column of the identity matrix one at a time, to get a matrix representation that can be passed to EIG.
If you increase the size of the input matrix, you'll see the 'StartVector' be passed in on the first call to the function handle:
n = 21;
A = ones(n);
StartVector = flip(1:n)';
[vec,ev] = eigs(@(vec)test_eigs(vec,A),n,1,'largestabs','MaxIterations',300,'StartVector',StartVector);