Not everywhere. Australia supports both the European pedelec classification of 250W with a 25km/h cutout and a local one of 200W with no speed limit (which can be hand-throttle in some states but must be pedal-assist in others). New Zealand just has a flat 300W with no speed limit.

I live out in the country in Australia. I’m planning a velomobile with an auxiliary motor for both touring and local use, and with this combination I expect and hope to attain an average speed of over 50km/h for the ~39km trip to the nearest town for church and groceries on Sundays. (I currently tend to average 22–24km/h on my recumbent tricycle with no motor, the trip typically taking 1h35–1h50m, with an estimated average power output of 120W. 200W is quite a lot to add!)

In the US, class 3 electric bicycles can assist up to 28 miles per hour, or around 45 km/h. I might be able to ride 28 miles per hour on a regular bicycle, but I can't do it for very long.

Are they regulated like non-electric bikes? In Europe, I believe the main difference is insurance and license, which are required for e-bikes with motor-supported speeds > 25km/h (at least in some countries, not sure how uniform it's handled).

As far as I know there is no jurisdiction in the US where you need either a license or auto insurance for a class 3 electric pedal assist bicycle. I think you generally ride them wherever regular bicycles are ridden.