[P1] cloud: Neon branch-aware connections

[sri-rang]

Summary

Neon's standout feature is instant database branching — copy-on-write branches for dev, preview, and test environments. Developers working with Neon accumulate many connections in Tusk (one per branch) with no indication that they belong to the same project. There is also no way to create a new branch from within Tusk.

User Story

As a developer using Neon, I want Tusk to group my connections by Neon project and let me create new branches without leaving the app, so that branch-based workflows feel native.

Acceptance Criteria

• Connections to Neon hostnames (*.neon.tech) are automatically grouped under their project name in the sidebar (detected from the hostname pattern <project>.neon.tech)
• Each Neon connection badge displays its branch name (parsed from the hostname or stored in the profile)
• A "New Branch" action in the connection group context menu opens a dialog to create a Neon branch via the Neon API, then automatically adds the new branch connection to Tusk
• Neon API key can be stored per-project in the connection settings; Tusk uses it for branch management operations only
• Branch connections indicate the parent branch name in a tooltip on the connection badge

Notes

• Neon connection hostnames follow: <endpoint-id>.neon.tech; the project and branch are encoded in the endpoint ID
• Neon Management API: https://console.neon.tech/api/v2 — requires a personal API key
• Branch creation API: POST /projects/{project_id}/branches
• This feature requires opt-in (the user must provide their Neon API key); connections without an API key still get grouping by hostname pattern