Is chess played by exactly the same rules around the whole world?
No. To be honest, as outlined below, even FIDE rated chess is played by slightly different rules because the FIDE Laws give this flexibility
According to the preface of the FIDE Laws of Chess:
A necessary condition for a game to be rated by FIDE is that it shall
be played according to the FIDE Laws of Chess. It is recommended that
competitive games not rated by FIDE be played according to the FIDE
Laws of Chess.
Only games which are going to be rated by FIDE have to be played by the same rules but there are cases where games are not going to be FIDE rated where it makes a lot of sense to depart from parts of the FIDE Laws.
One obvious example is children's chess. In December I was one of several arbiters in a large room where several children's tournaments were being played, with different strength players in each and slightly different rules regarding clocks and illegal moves. For the stronger players in the FIDE and nationally rated tournaments clocks were used and the "2nd illegal move loses" rule was strictly enforced. In the weaker, unrated tournament there were no clocks (unless a game looked like taking too long in which case a clock was introduced with 5 minutes per player) and a loose "3 illegal moves loses the game" rule.
The English Chess Federation publishes its tournament rules which outline some of the differences. This is what it says for non-FIDE rated games:
Section D – Optional rules for events that are not FIDE-rated
FIDE recognises for its own events and matches only one system of notation, the Algebraic System. For events that are not FIDE-rated, organisers may permit the use of descriptive notation.
For events that are not FIDE-rated, organisers may permit the use of adjudication.
For Under 11 Rapid Chess events that are not FIDE-rated – i.e. those restricted to players who are under the age of 11, defined by either the calendar year or the academic year – the number of illegal moves that results in the loss of a game may be increased from two so long as it is prescribed in advance in the regulations of the event.
Where no arbiter is appointed, e.g. in an evening league match, then the organiser is empowered to make decisions otherwise the responsibility of the arbiter with respect to Article 18.104.22.168. In the case of a team match, the organiser is also empowered to delegate this responsibility to the match captains.
Note that there are also areas of the FIDE Laws where there is a choice defined with a default. Here is how the ECF (see above document) handles one of the more contentious ones, the one which results in a zero time default rule if the organizer doesn't specify a more sensible time in advance:
According to Article 6.7.1 of the FIDE Laws of Chess, the regulations of an event shall specify a default time on the entry form. Should the organizer fail to do so, the default time shall be 30 minutes for a Standard-play game and 10 minutes for a Rapid Chess game. Blitz games shall have no default time – the time shall just be allowed to expire.
The document also usefully highlights other areas where FIDE gives options (which naturally give rise to even FIDE rated chess being played by slightly different rules in different tournaments):
Section C – Optional Rules
According to Article 10.1 of the FIDE Laws of Chess, the regulations of an event may specify a different scoring system. For example a player who wins his game, or wins by forfeit, scores three points (3), a player who draws his game scores two points (2), a player who loses his game scores one point (1), a player who loses by default scores zero points (0). If not specified, normal scoring is used (1, ½, 0).
Leaving the playing area:- According to Article 11.2.4 of the FIDE Laws of Chess, the regulations of an event may specify that the opponent of the player having the move must report to the arbiter when he wishes to leave the playing area. If this is not specified, there is no obligation for the opponent to communicate his intention to leave.
According to Article 11.10 of the FIDE Laws of Chess, the regulations of an event may specify that a player cannot appeal against any decision of the arbiter, if he has signed the scoresheet. If not specified, the player may appeal even after signing the scoresheet. It is strongly recommended that an Appeal Committee should be set up in advance.