Does attaching Boots of Swiftness count as "choosing a friendly unit" to trigger the effect of Irelia Blade Dancer?

Yes, attaching Boots of Swiftness to a friendly unit counts as "choosing" that unit, which will trigger the ability of Blade Dancer (Irelia's legend ability).

According to RiftJudge FAQ #7601 and #6203, equipping gear to a unit involves targeting that unit, and in the context of Irelia's mechanics, "choose" is equivalent to "target." Therefore, because the act of attaching gear requires you to choose a unit to attach it to, it satisfies the condition "When you choose a friendly unit" for Blade Dancer.