I have a design for a control looking like this:
I'd like to know if there is a commonly used name for such control to google its implementations.
As suggested on Kaylee Chatham's comment, it is also called multiple selection listbox or multiple choice listbox.
As seen on number 6 in this confusing page I found, this could also be called an Accumulator:
Description: When a user needs to select items from separate sets of search results or other large, unwieldy data sets- let them manipulate the left (search, sort, navigate or filter), and add to the right.
I would actually consider this more than one control, two list boxes and several buttons.
Using the terminology recommended by the Microsoft Manual of Style 4th edition, you would refer to the lists as "lists" within text.
In the Available Tasks list, select a task, and then click the the > button.
I would be leery to call it a "multiple selection listbox" because there are list boxes that allow users to select multiple items that are distinct from the controls shown in this question. If I needed a single word for this control, I'd probably call it "a double-paned list box", or something like that. But my preference would be to not call it anything and refer instead to individual parts. I think trying to force a name on this control as a whole would come off as jargony and not easily understood by a general user.