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Your zero cached_tokens results match the documented API boundary, not a missing agent setting. Microsoft lists Messages and token counting as Claude APIs in Foundry. Prompt caching is a Claude capability, but the Foundry Responses reference documents agent_reference without a cache control or agent/deployment cache switch. Caching prompt-agent instructions through /openai/v1/responses is therefore not a documented supported path.
For caching, call the deployment through the native Messages API, keep the reusable system prefix identical, and follow the native Messages caching contract. This requires sending the system prompt in the request instead of relying on server-side prompt-agent instructions. Verify operation from the native API's returned usage across repeated requests.
Claude models are supported for agents through Microsoft Agent Framework, but that does not imply prompt-caching support with agent_reference. Microsoft documentation publishes no roadmap or delivery date, so use Microsoft Support if you need an authoritative production commitment.