It looks like if both the tile movement objects are solid, then they will push each other based on the bounding box rather than the collision box and not block movement. 
Ashley is this intended behavior?
If you turn off solid for your object with default controls, they move together fine even with exact fit collision boxes.
To work around this, you might want to not use the built in solids behavior at all, and make your own pushing/blocking mechanic through events.
Hey 
oosyrag, thank you for the quick response. But I'm scared that I didn't describe the intended behavior fully and that could be a cause of misunderstanding. Yes, you are right. Everything works as expected when solid behavior is disabled, but then blocks are allowed to overlap each other.
I want to create a game slightly similar to Tetris, where a player can move different shapes, stack them together but do not overlap other shapes on the screen.