Although you don't want equal, I think there is a different word that is similar but a better fit.
My interpretation of this is based not on an actual company's organization or visual representation of employee roles, but on the underlying principles.
"Vertical is to hierarchical as horizontal is to egalitarian."
: asserting, promoting, or marked by egalitarianism
1 : a belief in human equality especially with respect to social, political, and economic affairs
2 : a social philosophy advocating the removal of inequalities among people
In other words, people at a company who have the same so-called rank are people who, as a group, are treated in a like fashion. No employee within a particular stratum is any different than any other employee in that same stratum.
Note that if you take hierarchical to mean a visual representation, rather than a principle, then while "vertical is to hierarchical," so is horizontal. Within a hierarchy, people will fit both vertically and horizontally.
If you really are talking about a visual representation, then a better analogy would be chess.
"Within a hierarchy, vertical is to rank as horizontal is to file."
3 d : any of the rows of squares that extend across a chessboard perpendicular to the files
file (noun 3)
2 : any of the rows of squares that extend across a chessboard from one player's side to the other player's side