Previously SpaceX Lunar mission date was 2022, but it is mentioned in the wiki that it is no earlier than 2023.
Why the date has been postponed?
The original lunar mission was proposed via Falcon Heavy and Dragon in 2018 and then delayed until it was cancelled. The Musk provided explanation is that the cost of man rating the Falcon 9 to NASA standards was worse than they expected, and for the number of Falcon Heavy flights, and the even smaller of likely manned flights, it was not worth the money. Especially since BFR was coming sooner rather than later, due to the long delays in Falcon Heavy.
Now that they have switched to using a BFR vehicle, they have to actually build, launch, and fully test it. This takes time. Per Elon Musk on the announcement of the first passenger, they had built the first barrel section (Need three or more for each of the BFR stages) so development is very early, and timeslines have stretched out.
This is rocket science.
A commentor noted that getting BFR man rated will be hard. Which of course it will, but notice the first manned customer who has paid a deposit is not NASA. Thus they really only need to satisfy the FAA about human safety, not NASA.
Rand Simberg has noted that it is possible that the NASA safety standards may not be as based on reality as they would like to portray, in fact he wrote an entire book on the topic of safety and the NASA approach.
The Previous SpaceX lunar mission was also planned with the same Japanese billionaire and on-board the falcon heavy and using the dragon capsule. However, SpaceXhas redirected the focus onto the new BFR project which seems to have finally fixed quite some design aspects aesthetically and are working on a 5 year time line to get it Human spaceflight ready. As the comments and the other answer points out. testing for human-centric spaceflight along with making test flights without humans can cost a lot of time and money. To ensure the safe return of the private individuals going up there without any technical staff maybe a challenge as well, so rigorous testing will be crucial and it will be a make/break point for the private space tourism industry that aims to offer tickets to space and make it as safe as possible. The 2023 already seems a highly optimistic target for the BFR.