In V13, the new DSolve option IncludeSingularSolutions -> True yields the sought-after solution:

DSolve[{y'[x] + 2 E^x y[x] - y[x]^2 == E^(2 x) + E^x, y'[0] == 1}, 
 y[x], x, IncludeSingularSolutions -> True]

DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution.

DSolve::bvnr: For some branches of the general solution, the given boundary conditions do not restrict the existing freedom in the general solution.

(*  {{y[x] -> E^x}}  *)

In V13.1, the new DSolve option IncludeSingularSolutions -> True yields the sought-after solution:

DSolve[{y'[x] + 2 E^x y[x] - y[x]^2 == E^(2 x) + E^x, y'[0] == 1}, 
 y[x], x, IncludeSingularSolutions -> True]

DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution.

DSolve::bvnr: For some branches of the general solution, the given boundary conditions do not restrict the existing freedom in the general solution.

(*  {{y[x] -> E^x}}  *)