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The code looks pretty straight-forward and I don't spot any obvious simplifications. I would suggest that you add error handling - especially for the case when uid is not null and remains that way. Perhaps an error message should be added if the request for the appropriate value fails.

Since ecmascript-6 is used (e.g. with the template literals) const could be used instead of var for any variable that doesn't get re-assigned (e.g. the components, db, etc.) and let for other variables (e.g. uid).

You could also consider providing a key attribute on the v-for where the messages are displayed, but the markup is quite simple for those so it may not add any benefit.

It is recommended to provide a key with v-for whenever possible, unless the iterated DOM content is simple, or you are intentionally relying on the default behavior for performance gains.¹

¹_{https://v2.vuejs.org/v2/guide/list.html#key}