Is it illegal to change a public Wi-Fi name if you are not the admin? (The router allows anyone to access the “admin” simply by typing the router IP.)

It could well be. In the US, doing something that interferes with someone accessing their data can be charged as a felony, with jurisdiction under the Department of Homeland Security.

If you change a public Wi-Fi name (SSID), you may well be interfering with someone accessing their data in the cloud, or on one of their servers. This could be viewed as terrorist activity.

Just speaking personally, I’d rather face a few hungry lions than the DHS.