Synth playing can be remarkably like playing piano or can be completely different.
- Keys can be weighted to feel like piano keys, or unweighted
- Synths can use attack, sustain and decay parameters that are similar to piano characteristics, or they can be entirely different (eg slow attack, variable sustain etc)
- Synths can be configured to play the notes you press, or trigger arpeggios, sound effects, harmonies, whatever
You say "Synth players press and hold a few keys and let the synth do the job mixing in some melody" - this is an incredibly limited view that is also wrong in most cases. A good synth player has most of the skills of a good piano player, but also can add to this with programming.
If you want to learn to play any kind of keyboard, in the beginner and intermediate stages, most of the finger exercises and practice are going to be the same.
The programming skills will require a synth to learn, however, so at some point you will need a synth or other MIDI-capable keyboard.