I don't subscribe (entirely) to the "Bond as code name" theory (although the anachronistic DB5 in Skyfall points that way), the ending of Spectre does fall neatly into this idea:
- Connery Bond leaves after You Only Live Twice;
- Lazenby Bond ("that never happened to the other guy") leaves when wife Tracey is killed in OHMSS;
- Connery Bond comes back in Diamonds are Forever (fatter, hairpiece, looks like he's been enjoying the high life for a few years) and leaves with Tiffany Case;
- Moore Bond leaves with Stacy Sutton in View to A Kill (and is 58!);
- Dalton Bond leaves with CIA agent Pam Bouvier after he has gone seriously rogue and lost his licence on a personal revenge mission in Licence to Kill;
- Brosnan Bond leaves with Jinx (and a big box of diamonds) in Die Another Day;
- And now Craig Bond (appears) to leave with Madeleine Swann and the rebuilt DB5.
Indeed it was Die Another Day that apparently wanted to legitimise the Bond as Code Name idea (for the 40th anniversary), we see lots of old gadgets, and it was rumoured they were trying to get Connery to cameo in the secret underground station. However someone got cold feet (or Connery said no) and the idea was shelved, but the ending of that is suggestive in the same way as Spectre, Bond and Jinx aren't "discovered" in the way Bond is normally caught with his conquest at the end of the film suggesting they never came back (especially when the next one has a younger, blonde Bond being promoted to 007)
UPDATE Oct 2021
Having now seen Craig's final outing in "No Time To Die" there seems to be a tacit admission of this theory. Partly due to the ending (which I won't ruin by discussing, but we all know this is Craig's final outing, right?), but mainly due to details in the film.
We see Craig Bond at Vesper Lynd's tomb, which lists her as having died in 2006 (so the films broadly follow with real time).
Bond uses the DB5 (Connery Era) which Q had rebuilt (with the mini guns as seen in the trailer), and later returns to London.
When he arrives he goes to ANOTHER lockup (ala SkyFall), and pulls the sheet off (wait for it), Timothy Dalton's Aston Martin V8 Vantage, complete with 1985 UK licence plate (see below)
Given the realtime aspect confirms the events of the Dalton films happened in the 80s, it follows they must be different Bonds, and there is an overall continuity.