So there doesn't seem to be a place in the bug tracker to report this which does not fill me with confidence, but eh, might as well try!
In the iOS app Don't Starve, the Virtual Stick does not block interactions with the objects behind it. However, in the Don't Starve Shipwrecked app, it does block, so touching the stick control does not cause any other action to occur -- the Shipwrecked behavior is the desired behavior.
For example, if you are in a dense forest at night and you are holding a torch as you run from spider -- a scenario we've all experienced, I'm sure -- if you let go of the stick control (say, to craft another torch) you can end up being unable to move again! You can get trapped in a cycle where you try to touch the stick control to begin moving, but underneath the stick is a flammable tree, and pressing on a flammable object while holding a torch opens a context menu to examine the object or light it on fire, and opening a context menu closes the stick control until a menu option is chosen, so your options become: stand stoic as you are eaten by spiders, unequip your torch so you can be sucker-punched by Charlie, or set everything on fire and laugh as it burns.
1. Start or continue a game. 2. If not already done, open settings and enable the virtual stick. 3. Move the character around until the center of the stick control is covering an object that the character can interact with (like a tree which will be examined when tapped). 4. Remove your finger from the virtual stick. 5. Place you finger back on the virtual stick and attempt to move the character. 6. The character will interact with the object, then move (unless the interaction opens a context menu, in which case the movement will be cancelled). Often the movement will cancel the attempted interaction if it required the character to move toward the item (e.g. if the object is a tree and the character has equipped an axe).