In RPB Chessboard, caching is used to store some dynamically generated CSS code, that is not likely to change often, but that cannot be defined in a fully static way. The idea is to avoid having to much CSS inlined each time a page is generated.

Do you have a suggestion to achieve this in a more proper way?

WordPress has its own built-in caching method. You can use wp_cache_set() / wp_cache_get() for caching.

Complete documentation is here:
https://codex.wordpress.org/Class_Reference/WP_Object_Cache

But WP_Object_Cache is not intended for file caching, am I wrong?

WP_Object_Cache is a key/value store. You can use it for many things. I've created a PR uses the object cache.

If you're concerned about inlining the CSS, I can rewrite this so we generate distinct, cacheable URLs for each cached CSS blob and enqueue them.

It seems that the function wp_add_inline_style( $handle, $data ) does nothing if wp_enqueue_style( $handle, ...) has not been called previously, with the same $handle. I've fixed your PR to account for this pitfall in the WP API...