This picture does not show an electric car charging in 1905. It's fourteen years off, but that's just the start of things.
This is indeed an old electric car charging, but it was taken in 1919, as the Library of Congress notes. The first tipoff is that the car is a Detroit Electric, which didn't start production until 1907. Here's the picture in full.
What's interesting is that there was electric car charging earlier in the 20th century, and even charging stations. General Electric showed off an 'electrant' (like an electric fire hydrant) for street charging in New York City all the way back in 1900, as a contemporary electrician's journal reports, and they kept making them at least through 1914.
The thing is, electric car charging around the turn of the century might look pretty modern in photographs, it was a much more involved process. Battery technology was way behind where we are now. Tons of corrosion needed regular cleaning and batteries themselves could easily be overcharged. You'd have to get a custom-built control panel, as the excellent book Taking Charge: the Electric Automobile In America explains. Some early electric car buyers took the step of hiring their own electrician.
So yes, we did have electric car charging over a century ago, it just didn't look quite as it does in this picture.
Photo Credit: Cress-Dale Photo Co., Seattle/Library of Congress