You can update every Rent-to-Own plugin (Except for Arturia plugins) via the Splice desktop app. This also applies to Rent-to-Own DAW's such as Studio One Professional 4 and 5. If there's an update available, you'll see this in the Apps & Plugins section of the Splice desktop app:
Click on "Update" and follow the on-screen prompts from the installer.
If there are no updates available for a Rent-to-Own plugin, you'll see the option to "Reinstall" the plugin:
Please note: The "Reinstall" button isn't commanding you to reinstall the plugin. It's giving you the option to reinstall if you find reinstallation is necessary.
Serum updates are a two-part process. First, you install the base version and then you install the latest version. If it's bringing up the older installer, run the installation and then press "Update" or "Reinstall" after in the Splice desktop app and run the update installation.
All updates for Arturia plugins are done directly through Arturia Software Center in the My Updates section of Arturia Software Center.