The "spacer" would also be referred to as a mount, bracket, bushing.
OEM ones are usually rubber and wear out over time.
The mount he is referring to could be on the transmission or the differential.
For example, mine has a "thud" sometimes when shifting gears coming from my differential (comes from the rear as mine is RWD, yours comes from the front as it is FWD). A polyurethane replacement for my car is about $160 plus labor involved of swapping in the new mount.
The purpose of these mounts are to hold the pieces in place while allowing for some movement to dampen noise and vibration from transferring to the chassis. With it worn out, there is a space between there and when force is put through the drivetrain, the parts move, and you get your "clank" as it hits.
He is right in it being "optional" but it may have side effects over the long term. It is allowing movement where there shouldn't be. This could cause other moving parts connected to wear out sooner.
For example, on a previous vehicle I had a bad motor mount (serves same purpose, mounts the engine to the chassis but dampens the vibration) and the additional movement of the engine eventually caused damage, it wasn't overnight though.
In general it shouldn't hurt your performance to any noticeable degree. If this was a drag car and you were trying to get every fraction of a second cut off, then sure it could. That moment where the force is causing the transmission/differential to shift and impact the chassis could have been spent going to the wheels. For daily driving purposes, it'd be negligible.