TLDR: The European Commission introduced its Fifth Anti-Money Laundering Directive (AMLD5) in 2018, which came into effect on January 10, 2020. While AMLD5 introduces licensing requirements for crypto exchange and wallets, it makes no mention of the Travel Rule.
The EU introduced AMLD5 on May 30, 2018, which added a requirement for member states to license Crypto to Fiat exchanges as well as Custodial Wallet services. It came into force on January 10, 2020, requiring member states to implement the directive. Individual member states are in the process of implementing these rules.
We have heard from customers throughout the EU that their national regulators have reached out to discuss the application of the Travel Rule. In general what we hear is that regulators want VASPs to start implementing the Travel Rule now, but are taking a best-effort approach initially.
AMLD5 does not include Crypto to Crypto exchanges as obliged entities, nor does it include the Travel Rule specifically. A new directive will probably include fulfilling the full FATF guidelines in 2020/2021.