When did the Royal Navy start to fill a given rank of admiral with more than one officer?

According to N.A.M Rodger the transition to multiple officers in the same post was (officially) in 1743,

There were eight flag officers' ranks: Admirals of the White and Blue, Vice-Admirals and Rear-Admirals of the Red, White and Blue. Until 1743 there was in principle only one officer of each of these ranks. By that time the Navy was acutely short of admirals, and as soon as George II was persuaded to allow a multiplication of flag officers, their numbers increased rapidly.

^{Commissioned officers' careers in the Royal Navy, 1690-1815, JMR, 3:1, p87}

There were no automatic appointments or promotions in the Royal Navy. Officers would only be appointed (commissioned) by order of the Admiralty. However, the Admiralty generally worked on the principle of promotion based on seniority. So when a post at flag rank became available (due to the death or incapacity of its previous holder), the most senior officer (at the rank below) on the seniority list usually got the job.

Seniority was based on the date that the officer obtained the rank of Post Captain. So it was advantageous, if your goal was to become an admiral, to become a post captain as young as possible. This resulted in many abuses that had children "serving" on a ship's books (while safely ashore) to gain the sea service required to pass the Lieutenant's exam, which was the only real hurdle to become a Post Captain, as young as possible.

One of the other problems with the seniority system was that, potentially, the most senior captain on the list might be too old to do the job and, consequently, there was a system of "superannuation" that effectively granted them the rank but kept them off the active list.

For example, in 1769 there were almost as many superannuated admirals as there were active ones.

Flag-ranks in 1769
The honorary rank of ‘Vice-admiral of Great Britain’ was held by Admiral of the Fleet Edward Hawke, 1st Baron Hawke (1705-1781), who was appointed in 1765.

1 Admiral of the Fleet: Edward Hawke, 1st Baron Hawke (1705-1781) 15 January 1768

Admirals (7)

3 Admirals of the White;
4 Admirals of the Blue

Vice-admirals (10)

3 Vice-admirals of the Red; 3 Vice-admirals of the White; 4 Vice-admirals of the Blue

Rear-admirals (11)

4 Rear-admirals of the Red;
4 Rear-admirals of the White;
5 Rear-admirals of the Blue

In 1769, there were also 22 rear-admirals who were considered ‘superannuated’ (incapacitated or disqualified for active duty by advanced age)

^{Promotion in the Flag Ranks in the Royal Navy during the Napoleonic Wars, Stephen Millar}

However, the Admiralty could be pragmatic when circumstances required...

The system was extremely inflexible in that it was impossible to overtake those above you on the list; but this did not stop the Admiralty appointing a man from well down the list to command a squadron or a fleet if he was felt to be ‘the best man for the job’. However, no flag officer could be placed under the orders of someone blow him on the list. The longer one had one’s commission, the higher one was up the list.

^{Promotion in the Flag Ranks in the Royal Navy during the Napoleonic Wars, Stephen Millar}

Some caution should be taken when looking at "snapshots" of the numbers of officers serving at any given point, based purely on the Admiralty lists.

These fragmentary figures raise more questions than they answer, and they conceal a critical problem. All of them are cast in terms of the total number of officers of a given rank at a given date, and the unwary reader might naturally assume that these were officers 'on the Active List', in modern parlance; available to serve. Unfortunately for the historian, there is no doubt that some officers remained on the official lists even though they were no longer fit to serve. An unknown, and probably variable proportion of the officers listed in the tables above, and in all other figures deriveable [sic] from the Admiralty's official lists, were not in practice available for service.

^{Commissioned officers' careers in the Royal Navy, 1690-1815, JMR, 3:1, p87}

Links:
Professor N.A.M. Rodger (2001): Commissioned officers' careers in the Royal Navy, 1690-1815,
Journal for Maritime Research, 3:1, 85-129

Promotion in the Flag Ranks in the Royal Navy during the Napoleonic Wars, Stephen Millar