It was not uncommon to find these "Exchange Office" registry labels applied on top of the stamp. These labels were used between 1883 - 1911 in conformance with a Universal Postal Union mandate.
There was no regulation or instruction that I have found that specified placement. For the cover mentioned, it was probably placed on the stamp as the writing on the cover left no clear blank space for the label. These were mostly printed on a translucent paper, so that the postal clerks could see most of what was underneath if necessary.
These were required on all mail from the United States to a foreign destination (Canada was an exception), and the label was applied at the post office where the letter exited the United States. They were also used on any foreign origin mail that passed through New York en route to another foreign country.
Cover 2 in this series should also have had a label present, but it appears to have been removed or perhaps fell off.
There were nearly 30 different Exchange offices which used these labels, most with their post office name printed on them, although New York dominated the registered foreign mail system. Many of these are quite scarce.
San Francisco, California was the Exchange Office for most mail to the Far East and Australia.
I posted a thread relating to Texas registered mail a few years back, which included a number of these covers, including some from other Exchange Offices located along the border between Texas and Mexico. It can be seen at:
https://goscf.com/t/48701You will need to scroll to the end of the thread to find the covers with Exchange Labels.