To what extent do academic promotions share similarities with promotions in industry?
They are really not very similar at all.

(This answer describes common structures at US universities.)

There are a number of very significant differences between "promotion" in academia and in industry.


The vast majority of faculty members will be promoted at most twice in their entire career.  Once from Assistant Professor to Associate Professor, and once from Associate Professor to Professor.  
Promotions cannot be granted unilaterally by a single manager.  Either of the two promotions mentioned above typically requires something like: a majority (or supermajority) vote of the faculty in the department, the recommendation of the department chair, the approval of a dean and a provost (and maybe a Promotions Committee), and often the university president and the board of trustees.  All of these are required for the promotion to take place.
Promotions cannot be sped up.  University policies usually require a fixed minimum (and sometimes maximum) number of years of service at one rank before being eligible for promotion.  In some cases a candidate may be able to negotiate to adjust these timelines at  the time of hiring, but usually not afterward.
Entering management isn't the "normal" path.  The Professor ranks described above are not really management positions, and the duties of each rank are roughly the same: teach, do research, engage in institutional service.  The most common "management" position for a professor would be department chair, and in many cases this is a fixed-term position, after which the professor resumes their previous duties.  A professor could also try to become a dean; but since this typically means the effective end of their research and teaching career (or at least a drastic reduction or long hiatus), it's not something they do lightly.
Management structures in academia are relatively flat.  A department chair may supervise up to 40-50 tenure-line faculty members or more, and perhaps a larger number of non-tenure-line faculty, postdocs, etc.  A dean may supervise a similar number of departments.  At my medium-sized university, there are only three levels of "middle management" between an ordinary assistant professor and the university president.  As a result, there are relatively few "management" positions available, compared to the number of "rank-and-file" professors.
Universities are to some extent run from the bottom rather than the top.  Significant decisions are often made (or recommended) by vote of the rank-and-file faculty, rather than by an executive; this concept is called "shared governance".  As such, management has less power than they would in a company, and professors don't need to move into management to have some influence on the overall direction of the institution.


As such, the sort of things that are commonly recommended when seeking promotion in business (showing value to the organization, currying favor with management, demonstrating ability to supervise others, etc) are not nearly as effective for advancement within academia.