Experiment tags (manual + auto-tagging) ⟵ foundational

Feature request

Is your feature request related to a problem?

Orgs grow fast and fall behind on experiment hygiene; large numbers of experiments go unorganized. A simple tagging system lets teams group/filter; auto-tagging on creation keeps it low-effort so process doesn't depend on discipline. We have tagging on FF, so should provide this as parity.

Describe the solution you'd like

When a user creates an experiment we should add the ability to add tags. We could do auto-tags, but question is whether the AI using the MCP can add these, or PHC. If we added an autotagger we could ensure tag consistency.

Describe alternatives you've considered

Additional context

Debug info

- [ ] PostHog Cloud, Debug information: [please copy/paste from https://us.posthog.com/settings/project-details#variables]
- [ ] PostHog Hobby self-hosted with `docker compose`, version/commit: [please provide]
- [ ] PostHog self-hosted with Kubernetes (deprecated, see [`Sunsetting Kubernetes support`](https://posthog.com/blog/sunsetting-helm-support-posthog)), version/commit: [please provide]

No files, tests, or entry points are identified in the issue. First clarify the experiment-tagging model and whether manual tags, automatic tags, or both are in scope, including whether tagging is handled by MCP or PHC; done should include an agreed creation flow and consistent tag behavior.