Does Irelia Legend target the unit it is readying?

No, the Blade Dancer (Irelia's legend) ability does not target or "choose" the unit it readies.

According to RiftJudge FAQ #220 and FAQ #703, the ability's effect of readying a unit is not considered "choosing" or "targeting." When you perform an action that triggers Blade Dancer (like targeting a friendly unit with a spell), the unit that was the subject of that action is automatically the one that Blade Dancer readies. Because no further choice is made by the player to determine which unit to ready, it does not count as targeting.

As a result:

• The Blade Dancer ability will not trigger any effects that specifically look for a unit being "chosen" or "targeted" (such as Irelia, Fervent's bonus for being chosen).
• However, successfully readying a unit via this ability will trigger effects that look for a unit being "readied."