No, the mixer does not need to be registered. Universal Control (includes UC Surface) is available on the download page (https://www.presonus.com/products/StudioLive-2442AI/downloads). This page lists all available versions of Universal Control and Capture. You need to check which firmware version is installed on your rmixer, because matching versions of Universal Control and firmware are a must.
If you want to update firmware, this is only possible in your PreSonus account (your mixer needs to be registered). Also downloads of Studio One are only possible through your account.