Do you know of any companies that handle the whole process, similar to how stripe handles the whole purchasing process?
Furthermore, even if you outsource the whole business process that handles sensitive data, you're still the 'controller' of the data, and you still carry full responsibility for it - you can and should have legal arrangements with the processor to recoup your damages/costs if they screw up, but you're the party that's directly liable, and if the processor can't/won't pay, that's solely your problem.
You can outsource certain parts of GDPR compliance - such as handling the paperwork and managing the customer enquiries properly, but it's hard(impossible?) to separate "the whole process" from the rest of your business.
What they do is somewhat similar to consulting and audit companies - they'll go over your internal processes and/or suggest standard procedures if you don't have any; they'll generally consult with the local data protection authority on particular interpretations and apply them to all their customers, etc. Creating/adapting a procedure for answering customer requests for their data (including the identity verification) would be part of the service; another common service is doing GDPR-policy training for e.g. call center employees. So a thing like that already exists; they won't take over your liability or your processes but they'll review your processes and hand-hold you through any adjustments needed.