Are the student solo endorsements a “package deal” with that one specific instructor?
The FAR's don't talk about whether or not a CFI can accept another instructor's solo endorsement. If the student has the endorsement, the 90 day period has not expired, and the make and model of the aircraft in the endorsement matches, then the student is technically good to go.

That said, a good CFI would carefully review the student's log book and, if necessary, require additional instruction/review before the allows the flight school to rent the student an aircraft. The relevant FAR is 61.87.