Why would the government collect Wi-Fi SSIDs via manual door-to-door questioning of citizens?
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).
This seems like a bureaucratic way of instilling FUD (Fear, Uncertainty and Doubt) in a population.
For example in the old Soviet Union, neighbors would routinely spy on neighbors and agencies collected notes on citizens. But the raw data of those notes were so enormous—and the bureaucracy was/is so inneffecient—that the actual information collected was utterly useless. But the fear of the data collection—in and of itself—kept people in line. As journalist Agnes Smedley once said:
“Everybody calls everybody a spy, secretly, in Russia, and everybody is under surveillance. You never feel safe.”
The whole idea is in an oppressive society nobody talks about spying, nobody confirms spying happens when it happens, but everyone is in fear about being spied on and knows one or two people who might have been a victim of spying so seeds of fear are planted. Thus these people behave as if they are spied on all the time no matter what they do. Which is to say they live in fear of the darkest aspects of the unknown.
So by bureaucrats doing what you describe, the government is acting in a way that reminds it’s citizens that they are watching you. And there is even the side effect of citizens themselves keeping tabs on each other such as stating some neighbor didn’t report an SSID and now they are being fined or their business is being hassled by local authorities. Never underestimate the power of a few neighborhood busy-bodies gossiping nonsense just to gossip.
If the goal of an oppressive government is to oppress, then random acts of questioning its citizenry can be considered one way of putting pressure on a a population to remind them who is in charge.
Because in a practical level, it is trivial for someone to drive or walk around a neighborhood and log all the SSIDs one can detect without bothering anyone. I mean Google does it all the time, right?