Yes a controller absolutely must listen to read-backs and ensure they are correct. This is commonly referred to as hear-backs.
Consider the following excerpt from ICAO Annex 11, Air Traffic Services (emphasis mine):
3.7.3 Read-back of clearances and safety-related information
188.8.131.52 The flight crew shall read back to the air traffic controller safety-related parts of ATC clearances and instructions which are transmitted by voice. The following items shall always be read back:
a) ATC route clearances;
b) clearances and instructions to enter, land on, take off from, hold short of, cross and backtrack on any runway; and
c) runway-in-use, altimeter settings, SSR codes, level instructions, heading and speed instructions and, whether issued by the controller or contained in ATIS broadcasts, transition levels.
184.108.40.206.1 Other clearances or instructions, including conditional clearances, shall be read back or acknowledged in a manner to clearly indicate that they have been understood and will be complied with.
220.127.116.11.2 The controller shall listen to the read-back to ascertain that the clearance or instruction has been correctly acknowledged by the flight crew and shall take immediate action to correct any discrepancies revealed by the read-back.
If, after a read-back by a flight crew, the controller neither acknowledges nor corrects the read-back, most flight crews will percieve this as an implicit confirmation of the read-back. This makes hear-back errors very dangerous, and is the reason why the pilot-controller communication loop has, not two, but three steps, the third step being the hearback:
While your example of an incorrect transponder code is rather trivial, incorrect read-backs can be extremely dangerous. For instance, they could result in a level bust, mid-air collision or runway incursion.
Therefore, performing appropriate hear-backs is an essential part of ATC training, and is taught to ATC trainees from day one. The importance of active listening is demonstrated in a simulator environment, where simulator pilots will occasionally - either intentionally or by mistake - make incorrect readbacks, the result of which will be a dangerous situation, if not corrected promptly.
Read-back or Hear-back