Widget Factories are born out of management school of thought revolving around motivation Theory X proposed by McGregor: rank employees are lazy and require constant control and supervision, jobs are held in the name of a pay check, managers are always able to do their subordinates' jobs to the higher or, at least, same standard. This thinking lands to a natural idea that the entire team can easily be replaced with and represented by the manager alone - after all everyone else on the team is either easilly replacable or there just to enhance manager's ability to complete tasks. Hence the hierarchy as a structure and rather horizontal job roles.
Widget Factory management operates on the assumption that software can be manufactured out of a specification prepared by a business analyst through a clearly defined process run under the close supervision of a project manager. The manufacturing is taken care of by staffing the project with enough qualified yet interchangeable programming and testing resources. Work is driven by a prearranged budget based on the initial business case prepared by PM and BA.
Management that runs a Widget Factory is easy to spot just by paying attention to the way these people talk. They are likely to be on about resources (including when referring to team members), processes, operating efficiency, uniformity, repeatability, strict control over use of resources, clear-cut job roles and defined process inputs and outputs. They'd casually mention the actual factory metaphor when trying to convey the image of the ideal software development operation as they see it.
Now, it won’t come as a surprise that most in-house software development teams and some consultancies are run as Widget Factories relying on a process to produce consistently boring software.