You have two important metric groups in nearly every piece ever written in 6/8:
1. The full measure divided into 6 eighth-notes, as you point out
2. Those six eighth-notes divided into two groups, each containing three eighth-notes
Musicians end up counting it something like this: "ONE two three TWO two three," with a primary emphasis on "ONE" and an ancillary emphasis on "TWO." These two accents are sometimes called the "big beats," or are notated as dotted quarter-notes both in the score and in the tempo marking. The accent pattern can also be described as "feeling it in two," meaning that the player picks a tempo suitable to make the big beats clear to the listener.
While there are some slower 6/8 pieces where the tempo marking might be specified with an eighth-note, the piece is still notated using dotted quarter-notes to mark the "big beats," and (aside from some temporary syncopations) the players still divide up the measures into two big groups.
It is counted and notated this way because-- ignoring temporary syncopations-- music written in 6/8 predominantly obeys those metric groups.
In other words, if you try to speak the eighth-note tempo in 6/8 doing, "ONE and TWO and THREE and," you will be going against the grain of the music. E.g., you'll be speaking that rhythm while the bass pattern might be articulating bass notes on eighth-note #1 and #4, and/or the melody follows a pattern in groups of three eighth-notes, and/or the middle-range accompanimental chords play on eighth-note #2, #3, #5, and #6, and so on...