What aircraft broke the most [total] world records?
I'm not exactly sure what type of recored you are looking for (civil, military etc.), but a specially modified Su-27 claimed 41 records for altitude and rate of climb between 1986 and 1990. As globalsecurity.org notes:

Between 1986 and 1990, using a specially configured prototype aeroplane T10-15, which became known as P-42, the design bureau's test pilots established 41 IAF-registered world records of rate-of-climb and flight altitude, some of the records being absolute.


P-42 Streak Flanker; image from su27flankerfamily.wordpress.com
Some of these records are given here,

On October 27 and November 15, OKB test pilot Viktor Pugachev added eight climb-to-altitude time records to the table of world achievements in just in two flights-four absolute records for land based aircraft and four records for aircraft with takeoff weight up to 35,274 lb (16,000 kg). Pugachev climbed to 9,843 ft (3,000 m) in 15.573 seconds and to 19,685 ft (6,000 m) in 37.05 seconds.
On March 10, 1987, and on March 23, 1988, test pilot Nikolai Sadovnikov flew the Sukhoi P-42 to 25,528 ft (9,000 m) in 44 seconds, 39,370 ft (12,000 m) in 55.2 seconds and 49,213 ft (15,000 m) in 70.329 seconds.
On June 10, 1987, Sadovnikov claimed a world record for a N category level flight altitude of 63,435 ft (19,335 m).

In some other publications, a number of 27 records have been reported; in either case, it is quite impressive.