I see two possible uses of such information from a government perspective. None of them involves the password or actually using your WiFi access.
Forensic analysis: connected devices store an history of access points they were connected to, sometimes associated with "last seen" dates. Using this history, it is therefore possible to know where someone was and when, which can be very helpful for investigators.
Concrete example: someone is arrested, his cellphone and laptops are seized for investigation, and their WiFi history is analysed (actually, in some cases, with some devices being a bit too talkative, it is not even needed to actually seize the device, but let's stay on topic). This will reveal where the suspect has been and when (for the last time at least), and because we are talking about associated access points it strongly leads toward some sort of relationship between the suspect and the AP owner (you do not distribute your WiFi password to any strangers, do you?), helping to construct a map of the suspect relationships (here having the ability to associate an SSID (the WiFi name) to an owner name takes all his importance).
Geolocation: If by any means investigators can remotely access the list of the access points covering the area where a device is currently located, then it is possible to determine where the device (and most likely its bearer too) is located.
Concrete example: An implant (to borrow NSA's terminology) is installed on a device with Internet but no GPS capability (laptop, tablet, etc.) or where the user has disabled GPS geolocation for privacy purposes. The implant phones home on a regular basis, sending a list of currently visible WiFi networks with the associated signal strength (the device doesn't need to be associated to any of them). Associated to a map of SSID geographical locations, this effectively allows to track in real time the suspect's movements.
In this case however, collecting the owner's name in such visible actions is less needed, war drivers and other Google cars know this very well. However depending on the details of this procedure it may also limit the possibilities for people to freely change their WiFi SSID name (let's say the form forbid this, it would be trivial for the authorities to detect undeclared changes and associate it to a name), thus possibly providing more accurate information on the long-term.
Regarding your mention about the WiFi password, as long as the WiFi access has been hacked by finding the password and not due to another unrelated weakness and unless the attacker also hacked the access point itself (and replaced its firmware for instance), then changing the password by a stronger one is sufficient to block any further exploitation of this access.
Regarding what can be done using a compromised access point, this is worth a separate question but you may already find a lot of information in already existing posts on this site (basically an attacker would gain a Man-in-the-middle (MiTM) position to intercept/modify all of your communication, this also opens opportunities to attack other devices of your internal network, and depending on the device's reset abilities the attacker could also prevent the access point firmware from being cleaned, effectively requiring the device to be replaced).
And yes technically you could change your WiFi "name" any time you want, however it is possible that your government may request you to fill a form to officially declare this change (or they just assume that only a minority of users will do this so it does not worth to track such changes).