Call ESPHome native API via Arduino

**Describe the problem you have/What new integration you would like**

It's quite easy to integrate Arduino libraries and run code via lambdas. But it would be nice to also have it the other way around.

I have an extensive Arduino project to which I would like to add Home Assistant features. (publish sensor data, perhaps expose a switch or service)

I know MQTT exists. But it would be really nice if I could just call the native API from my project.

**Please describe your use case for this integration and alternatives you've tried:**

MQTT. But the words from https://esphome.io/components/api.html?highlight=api#advantages-over-mqtt are very true.

**Additional context**

Start with the linked ESPHome native API documentation and compare its capabilities with the requested Arduino use case. Define the integration boundary and supported publishing, switch, or service operations; done would mean an Arduino project can call the native API without relying on MQTT.

Written by the indexing model from the issue text.