Aries leverages, Decentralized Identifiers[1] with Verifiable Credentials [2]
the "ledger" is where the public keys are stored. eg it could be a permission-ed ledger, similar to SSL certs only known/<want to be known> parties would publish their keys to the chain.
Example: Sovrin network [3]
Or could be permission-less, maybe focused more towards IoT/whatever
Example: ION Network [4] coming out of Microsoft.
The ledger is not a requirement to establish an identity as shown with the did:peer [5] method
Frankly I think the usage of blockchain was more to get on the marketing bandwagon at the time. Messaging is now moving from "blockchain" to "distributed ledger"
1 - https://www.w3.org/TR/did-core/
2 - https://www.w3.org/TR/vc-data-model/
4 - https://identity.foundation/ion/
5 - https://identity.foundation/peer-did-method-spec/index.html