What FAA stamps.....sorry ol' bean but after the bit of unpleasntness in the 1770's which resulted partly from a thing called the Stamp Act....our FAA does not use stamps except for postage and then it is actually metered or franked.
And....alas chappie....our Chief Pilots do not own rubber stamps for certifying logbooks.....antique procedure.....doesn't carry any water here with the officials.
The FAA will on occasion compare the entries one makes when applying for new certificates and privileges (think licenses.....) and any glaring contradictions may result in an audit.
In general, we use a self certification system and entries into the pilot logbook for required items such as proficency checks, Bi-ennial Flight Reviews, Practical tests, and such.......but no rubber stamps needed, Nigel.
Whereas you folks have the fancy booklet....with all the certifications, proclamations, and documentation.....we on the western end of the salt water divide.....use the logbook for such entries, carry our credit card sized Certificates (Airman and Medical),(once again...think license), in our wallet along with our money. The only thing we have to take out of that wallet when visiting the FAA is the FAA issued items.....the money we keep.....unlike your CAA.