Does Improved Knockback
When you successfully Shove a creature, increase both the distance you can push the creature and the distance you can move to follow along with the target by 5 feet on a success or 10 feet on a critical success. If you push the target into an obstacle, it takes bludgeoning damage equal to 6 plus your Strength modifier, or 8 plus your Strength modifier if you have legendary proficiency in Athletics.
apply to abilities like Tiger Slash with the "as if" wording?
You make a fierce swipe with both hands. Make a tiger claw Strike. It deals two extra weapon damage dice (three extra dice if you’re 14th level or higher), and you can push the target 5 feet away as if you had successfully Shoved them. If the attack is a critical success and deals damage, add your Strength modifier to the persistent bleed damage from your tiger claw.
I'd say yes.
The feat could have left that clause out and have you simply push the target 5'. Adding it in means they want it act like you Shoved them, and if you shoving them has extra perks, so would this since it's supposed to mirror that.