I am using Hashicorp Vault to store multiple secrets in the KV Secrets engine, one of which is the database connection string - username, password, host ip and port. I have multiple microservices, which need to use this db secret to connect with db.
Please clarify which of these integration pattern is valid:
The other microservice is an abstraction layer. It is extra work that might allow you to change secrets provider in the future.
Unless you can justify writing and maintaining that abstraction layer (because you want to use Vault in some deployments and AWS Secrets Manager in others), then don't bother.
The other issue is that although Vault's KV store is quite common and there are several other implementation, what if you want to use Transit, PKI or SSH CA? These services exist elsewhere (in AWS for example), but they don't have feature parity. You probably don't want to be on the hook to support those differences in your abstraction layer.
A low(er) cost alternative that allows you decouple the implementation from your code would be to wrap the Vault API class using a simple KVSecrets class in your code, a software design pattern know as the facade. But remember that unless you test your class with two services, you can't garantee it will be possible to migrate to another service one day.
So considering all this, just call the API directly or use the Vault library for your programming language.