This isn't an accurate take either. You can load client skills, sure, but the whole selling point from Anthropic is that skills are like memory, managed at the API layer "transparently" and that those skills are cross project. If you forced Claude to only use skills from the current project to try and be transparent, it would still be a black box that makes it harder to debug failures, and it'd still be agent and vendor specific rather than more human friendly and vendor agnostic.
I'm talking about skills as they are used in Claude Code running on my laptop. Are you talking about skills as they are used by the https://claude.ai consumer app?
Skills are just lazy loaded prompts. The system prompt will include the summary of every skill, and if the LLM decides it want to use a specific skill the details of that skill will be added to the system prompt. It's just markdown documents all the way down.
