I call myself a rock-hound when checking the shapes, patterns of lines and cracks, and different types of rocks compressed together on Manitoulin Islands North West shoreline. Glacial movement of years gone by created very unique and different sized stones and boulders here. Its a rock-hounds picnic area.
Beach stones are still a type of Rock. Someone who just collects rocks as a hobby can only be called a "Rock Collector". You could get a little more imaginative and descriptive and say "Beach Rock collector".
Someone who collects everything would be called a hoarder - this type of person may have OCD as well.
Beach
Some of the names given for a person who collects trains:model train collectormodel train enthusiasttrain modeler
Collision of the debris is the term that is given to the collision of the stones.
Weird
Quodophile.
Phonographer.
Gun-collector
There does not appear to be a name for it.
Arachnid collector or Arachnophile.
A Pottea.
simple - eraser collector!