Jim S asks an interesting question. Can ATC ask you to do something which is contrary to the regulations? The answer is they can, and just because they do, does not change the regulations.
The practice of advising aircraft to switch to advisory frequency and squawk VFR is common here, in Class C airspace, and I have experienced it in Class B airspace as well.
Discussing this with the ATC supervisor once, it was explained to me this way: The controller knows where you are going, knows your intent, and believes that he can manage the traffic in the area effectively if you leave the frequency. So for the controller it is an issue of risk and workload management.
With students, this creates a good learning opportunity to have them "monitor" one radio, while using another. No official guidance on this, but it is common practice. So if you are not familiar with this, have an experienced pilot or CFI demo it to you.
Keep in mind that this is the same process for IFR aircraft making approaches to non-towered airports.
Arguably, ATC can authorize deviations from 91.130, as @DaveCFII points out. Certainly that covers temporary frequency changes to check WX and things like that, but it could be interpreted to have broader scope is that it could be used to terminate communications near to the point where an aircraft leaves Class C.
Having said this, I would prefer that ATC state something like, "communications deviation approved." Of course that will not happen on a busy frequency.
Having said all this, it is important to remember that the pilot has final authority over a flight, and the FAA in the Granby LOI makes it clear that ATC does not have the requirement to point out to pilots what regulations apply to their operation. So a pilot who is uncomfortable with this procedure, might monitor ATC on one radio, and broadcast their intents on the advisory frequency. This way they can fully comply with the regulations. Most aircraft have dual COM radios. If not, delay your frequency change until you are clear of Class C, to remain in strict compliance.
Ref: Granby LOI https://www.faa.gov/about/office_org/headquarters_offices/agc/practice_areas/regulations/interpretations/data/interps/2006/granby%20-%20(2006)%20legal%20interpretation.pdf