The S3 API isn't unknown or anything, the client library SDKs all being open source. So I'd imagine a software developer writing a tool that aims to be S3 API-compliant would use one of the open source SDKs, and write their API while making requests to it locally through a client from one of the Amazon SDKs. Not trivial effort, but also pretty easy to imagine how you'd start off. If a client function from the SDK doesn't work with your API, you write your API handler to handle the HTTP request that function makes until it is supported.

I have wondered if Amazon has some additional tooling for other software providers to make their own S3-compliant APIs, but I don't know what Amazon's motivation would be to help make it easier for people to switch between other vendors. Whereas the incentive is much more obvious for other software vendors to make their own APIs S3-compliant. So I've so far imagined it is a similar process to how I described above, instead.

The S3 API reference [1] is closest to a formal spec there is. The request, response and the error codes are pretty well documented.

[1]: https://docs.aws.amazon.com/AmazonS3/latest/API/API_Operatio...