In practice not. Many e-bikes are capable of continuously operating at over 250 watts output, some at double or more that. They need to be, in the days when the power limit was observed much more strictly, e-bikes were frankly useless. This was why they made no market progress for two decades, a prospect only had to try one to realise they didn't need it.
What e-bike is it that you have by the way?
It's an Oxygen Emate City which shows the watts I am using. I very rarely use the top setting which gives me the 570w reading except for the odd 30 second burst on a very steep hill. It would also eat up the battery so it is used very conservatively.