I think I heard of something like this and it was due to the different sides of the road being under different Councils. Sounds silly, but it does work like that sometimes.
yes,it can happen anywhere as council tax bands were based yrs ago on the price of the house at that time,so you could have a 5 bedroom house next to a 2 bed bungalow in the same street,etc,etc,
Yes, my house was originally banded E, but other identical houses on the same street were a D. This was put down to an 'administrative error' when the banding was done. Essentially, they could not be bothered to visit the site to ascertain the types of houses on the street and lumped my house in with a load of similar larger houses on my street.
If they had both had an extension then one had been sold sometimes the council tax band then goes up when the new owner takes possession as it has 'bettered' but the old one would stay the same until it was sold.