There's a lot of confusion regarding this.

TLDR:
This actually involves two pieces of legislation:
- Travel Rule
- unhosted wallet verification

Both are Anti Money Laundering (AML) laws and not Know Your Customer (KYC) laws. On any exchange you have to tell the exchange who you are.

The first is global FATF[1] wrote a 'recommendation' and all jurisdiction on the world are expected to implement it. It mandates the exchange of originator/beneficiary data. When you do a regular/fiat bank transfer you also need to fill out the name of the beneficiary. This legislation brings that to crypto.

The second much less global. The Netherlands wants it, but the EU and UK (to a degree) will follow suite. The reasoning behind this law is that IF you DO NOT provider beneficiary data THEN you must be transferring to an unhosted/private wallet. You need to prove that you do.

Coinbase is merely a trailblazer here. Every single exchange will do this in the near future (within 2 years).

Elaboration:

This move has been a long time coming. FATF published their recommendation in 2019 and ever since the industry has been working to come of with a solution. There are MANY different approaches: TRP[2], TRISA[3], TRUST[4], Netki[5], VerifyVASP[6], Shyft[7], Sygna[8]. Some are purely a protocol, others are purely a SaaS offering and anything in between.

The main trouble with these laws is that exchanges need to exchange data. That means they need to talk to each other via mutually agreed upon standards. Added difficultly is that some exchanges are in a jurisdiction that requires this, while their counterparty is in a jurisdiction that doesn't require this. Solutions to this and more need to be devised.

Coinbase is heavily involved, and one of the initiators of TRUST. They are pushing this more than anyone else and see being compliant early as a competitive advantage. Binance, for example, got a reprimand of the Dutch National Bank. Making it harder for them to do business there.

While I'm not a fan of this regulation I believe this to be a crucial step into making Bitcoin (and others) mainstream. Without clear regulation institutionals won't enter. Retail will see this as a clear sign of endorsement from the government. The challenge is now to make the implementation of these new laws as sensible as possible, preserving privacy. Some initiatives I listed do that better than others.

The EU will have legislation in place in 24 months. The UK will start enforcing this in September 2023. Switzerland and Liechtenstein already enforce this since a couple of years. Singapore started to enforce this since not too long ago. As does South Korea. It is not clear to me when the US will actually start to enforce.

Disclaimer: I am one of the co-founders of 21 Analytics[9] and we are trying to make a living keeping the Travel Rule compliance space sane and clear.

[1]: https://en.wikipedia.org/wiki/Financial_Action_Task_Force
[2]: https://www.travelruleprotocol.org/
[3]: https://trisa.io/
[4]: https://www.coinbase.com/travelrule
[5]: https://www.netki.com/
[6]: https://verifyvasp.com/
[7]: https://www.veriscope.network/
[8]: https://www.sygna.io/
[9]: https://21analytics.ch/