OTOH, I have heard it used as a busy general aviation airport in the Denver area.

Not quite. In theory, "cleared to land" also "accomplishes exactly the same thing "

While, yes, you are always able abort a takeoff or perform a go-around on landing, one method involves ATC being on the same page during a training exercise, always a good thing. Arguably =especially= at a busy airport.

If it's used at my busy GA airport and not your busy GA airport, that's just one of the many local differences we find.

ATC could have used "runway 27 approved as requested".

2-1-18. OPERATIONAL REQUESTS
Respond to a request from another controller, a pilot
or vehicle operator by one of the following verbal
means:
a. Restate the request in complete or abbreviated
terms followed by the word “APPROVED.” The
phraseology “APPROVED AS REQUESTED” may
be substituted in lieu of a lengthy readback.
PHRASEOLOGY-
(Requested operation) APPROVED.
or
APPROVED AS REQUESTED

The big issue here is that the pilot requested an operation that will involve getting on the runway, starting a takeoff roll, and then stopping on the runway. The controller issued a clearance that would normally be issued to a landing aircraft. Should raise questions. Is ATC even talking to you? The controller may have mistakenly used the wrong callsign, and is actually clearing another aircraft for "the option". I personally wouldn't move until we(ATC and pilot/crew) came to a mutual understanding of the requested operation and ATC used some sort of verbiage that assurred me they understand and are approving my possibly lengthy occupancy of the runway. Opportunity to help educate a controller and your student, prevent a pilot deviation or avoid a real close look at the bottom of an aircraft executing the "option". Having said all of that I have heard it used many times just as you described. After 26 years of ATC I have also heard "what the hell is that guy doing", "I didn't clear him to for that", "Call the tower, advise ready to copy the number".

Local differences can't be contradictory with the ATC bible, 7110.65.

Yes, cleared for takeoff or for an airborne plane, cleared to land would work. But, there is already explicit phraseology for the latter, "cleared for the option".

Like many things in life, just because somebody does, does not do something, doesn't mean everything is fine and dandy.

As I stated, the best method is to contact the tower ahead of time to spell out your request !!!!

Tony
former ATC tower, approach, and center

I have heard ATC use the phrase "Cleared for the takeoff option, Runway 27" many times at my local airport for the purpose of training. The original requests sound something like: " XXX Tower, Twin Cessna XXXXX holding short of runway 27, requesting the takeoff option". This was common practice here until about a year or two ago. They will no longer issue a "takeoff option" clearance. However, the training is still accomplished by stating that the "takeoff will be aborted". For instance, "XXX Tower, Twin Cessna XXXXX holding short of runway 27, ready for takeoff. This will be an aborted takeoff." Or something similar to that. I think the main reason is so ATC has clear knowledge of the intentions of the departing aircraft in order to maintain proper spacing.

The takeoff option is commonly used at the airport where my former large part 141 flight school is based, as well. It may not be legal, but it sure is in common practice.

I've used it while at Purdue, and while I was an instructor in Michigan, at PTK and YIP. Basically I use it for a take-off roll, rotation, and low-altitude abort on the long runways. This was of course in SEL's like PA28's, C152's and C172's, no twins.

In fact, as an instructor, I would split the radio's if able(or I would isolate the student and they had no idea), and let the ground-controller know what my intentions were, and if they could pass it on to the tower controller. They would give the "option" clearance to verify what I had requested, and the student was none the wiser.