What owners control
Branding decides how the bot presents itself inside your server: the public name used in embeds and dashboards, the bot display name, the embed color, and the support link members are pointed to.
The dashboard always shows the resolved values next to the fields, so you can see what members currently see before and after a change.
Where branding appears
Saved branding is used by bot embeds, dashboard previews, support links, and message delivery names. It does not change your Discord server name, your roles, or anything Discord itself owns.
Changing the bot display name depends on Discord accepting the update. If Discord rejects it, the dashboard says so and keeps the rest of the branding saved.
Availability
Branding is a paid capability. On a package that does not include it, the controls stay visible but locked and the server keeps the default platform identity — nothing silently changes when a package ends.