It uses cell broadcast, not SMS, although the message might enter the network as an SMS before being converted to cell broadcast.
The way this originally worked was that you would manually input a list of topic or channel numbers you were interested in receiving messages about. Phones had a menu option for receiving a topic index, which was itself a broadcast message that listed which other topics were available locally.
I think perhaps the idea in the beginning was that there would be services like local traffic alerts, stocks and shares, news headlines, etc, that people could choose to subscribe to, but it never really took off. (I don’t imagine there was any profit in it.) Your phone number is not used to receive these kinds of messages; it is more like tuning into a digital television channel than receiving an SMS message.
It seems that all they’ve done in more recent phones is hard-code some emergency alert topic numbers that the phone will always be “tuned to” by default, displaying the resulting messages in a distinctive manner.
I imagine that most phones no longer expose a manual topic subscription list, because it wasn’t very useful, and those that do expose a manual topic list probably perform a boolean union between the manual topic list and the special emergency topics that have been enabled in the settings, so that those topics can only be disabled through the settings, or cannot be disabled at all.
For the US Presidential alerts, I wonder if there is any difference between what the law requires for carrier-provided phones and what it requires for phones that the customer purchases separately and brings to the network.