There are enough different names for the same concept that I won't claim to know what a Y-Y starter actually _is_. Here is a guess
In a star-delta starter, you don't actually reduce the voltage supplied to the motor at start. Rather you change the way the motor windings are connected together so that the current drawn by the motor is reduced. The voltage 'seen' by each winding _is_ reduced because of the different connection. The supply voltage remains constant, but there is a 1/root(3) change in coil voltage in star versus delta.
As an analogy, consider a lamp 'dimmer' made with switches. You have two lamps; in one switch configuration the lamps are in parallel across the supply, seeing the full supply voltage; in the other the lamps are in series. The supply voltage remains constant, but each lamp now 'sees' half the supply voltage.
Given this, there is no theoretical reason why a motor starter could not incorporate some different switching arrangement than star/delta, one that (say) reconnects a motor between 2 series star and 2 parallel star. This would give you a 1/2 change in coil voltage. With a complex enough switching arrangement, one could step through different connections and 'sneak up' on full voltage.
I've never seen such switching arrangements used. A 'soft start' probably makes more sense