The Bubbloid 554 is originally called Bubbloid112. It is designed by Carl Hoff and is the result of a collaboration between Cubewerkz and Calvin's Puzzles. Bubbloid puzzles are actually corner turning Cuboids that does not shape shift.
Each number on the original naming counts the number of shared edges between 2 adjacent corners sharing that edge.
For example, on the big square face, 1 edge pieces are shared between 2 adjacent corners on the square face. On the shorter edge, 2 edge pieces is shared between 2 adjacent corners.
So that is how the number 112 comes about. To make it easier, the naming is change to count the corner and edge pieces of the 3 sides. The new name Bubboid 554 is thus easier for layman to remember.
The mechanism do not have much corner-cutting feature that you find in speedcubes. They do turn acceptable, fun and challenge to solve. With some lube performance is much better.