We could add more context to org.hibernate.search.exception.ErrorContext; specifically we could add:

• String getIndexManagerName();
• String getShardIdentifier();

The shardname method is redundant as it could be inferred from the IndexManager name, but I guess it could be useful. Would these help for your use case?
Would you rather need a reference to the IndexManager instance, or a reference to other backend components?

I'm not sure what you mean by "It's not possible to compose multiple error handlers into one". Is something preventing you from creating a composite implementation, or is it that you'd expect to be able to provide multiple of them and have the search-engine facilitate the composition?