Does Irelia Legend count as choosing also?

No, using the Irelia Legend's ability (Blade Dancer) to ready a unit does not count as "choosing" that unit.

According to RiftJudge FAQ #6528 and #8590:

• The Irelia legend ability "Blade Dancer" does not target or choose a unit when you pay [A] to ready it.
• Because the ability does not target or choose, it does not trigger any other abilities that specifically require a unit to be "chosen" (such as the bonus on Irelia, Fervent).

However, Irelia, Fervent has a specific ability that triggers when she is either "chosen or readied." So, while the Legend ability does not count as "choosing" her, it does count as "readying" her, which will still trigger her ability to gain +1 Might.