Copyright © University of Cambridge. All rights reserved.

We have had a solution from someone who unfortunately didn't give his or her name, but it says that there are $14$ different rectangles:

1 |
2 |
3 |
4 |
5 |

6 |
7 |
8 |
9 |
10 |

11 |
12 |
13 |
14 |

Rectangles 1 and 4 are the same as each other.

Rectangles 5, 8, 10 and 12 are the same as each other.

Rectangles 6 and 11 (which are squares) are the same as each
other.

Rectangles 3 and 13 are also squares but they are not the same as
each other. They are similar to 6 and 11.

Rectangles 2, 7, 9 and 14 are different to each other and not
similar to any other rectangle.