The world has definitely gone mad. So it's the lights that I can't take a video of. So can I ask them to to turn the lights off so that I can take a video of the tower that I'm allowed to take a video of? In fact can't I sue the light company for hampering my ability to take a video of the tower at night.

Was this squarespace advertisement or actual info about Eifel tower and copyrights.

There is a little problem here. It's an old law that was revised almost the next week by the senate.
There was a campaign of false information published through wikimedia that turned viral with the first draft of the law.
You can find many pics of the "Tour Effeil" by night, illumination or not, in google on the first page with no other filter nor copyright.

You have the right to take pics of any architectural monument and sculptures you want, since it's not directly or indirectly intended for a commercial purpose.
So if you're an individual or an association, you can take all the pics of the Eiffel tower you want and share them.

If you're a private company or if you mean to sell them, you should in theory add a copyright if it's taken illuminated (night or day).
To be honest : no one cares. I never heard of someone who got sued for not adding "SETE – illuminations Pierre Bideau" on the pic.
The company in charge of the tower (SETE) even reacted to this by encouraging people to take pics by night of the tower

Moreover, there are challenges organized by state of Paris every years for the best pics of the city, there's no copyright on them but sometimes the name of the photograph.

So don't worry, take the pics you want and enjoy the city

If you are a private person taking private pictures for private and non commercial use, you can ignore everything in this video.

Im offended by the Eiffel Tower at Night, yuk