feat(app): show server name on session tabs - #51345

The following comment was made by an LLM, it may be inaccurate:

I liked your solution — it's much better than mine.

I opened #58155 before I found this one: it grouped sessions by server (Home and the vertical tab strip) instead of labelling each tab. Closing it so the discussion stays in one place. Your approach is much smaller and solves the same confusion, and labels work in the horizontal strip too.

If grouping is ever wanted on top of this, the branch stays on my fork.

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