Learning to visualize the board I have read in several sources, it's a important to be able instantly say what colour a square has or for example, if two pieces are on the same diagonal/rank/file, if their squares are given. And this skill can be learned by specific exercise for visualisation. 

Is this true?

Is it important to learn this skill isolated from other chess improvement learning?
This is not specifically for diagonals, but this is how I learned to visualize the board. I am not sure that I agree that you should be able to know what color a particular square is. I have been playing tournaments for 40 years this year, more than 30 at the master level, and I still do not think about the color of squares at all with regards to calculation (sure when it comes to making my bishop bad, and other positional elements). I just know where my pieces are, and with experience, where they can go. If I can visualize the former, then the latter is easy.

This is related to tactics, but specifically meant to be different. You can use a different book, but I started with Chernev's "Combinations, the Heart of Chess". The reason I used this book is that many of the examples are longer than normal, but not full game length, and they are interesting and fun. Some get to be 20 moves, or more.

Except maybe on short ones, you are not trying at all to solve them. ALL you are trying to do is keep the position in your head accurately, including sub-variations that are given. When you have visualized it well enough that you think you have it right, put it on a board, and see how well you really did visualize the position, especially, the final position. As you get better at this, you can use full-length games. Again, unlike tactics training, you are just trying to visualize where the pieces are.

Another fun book to do this with is Chernev's "1000 Best Short Games of Chess" since it starts with 4 moves and gradually goes up to 25 max. The only problem is both books I mentioned used descriptive notation, and if that is a problem, you will need to find other interesting shortish games to do this with.

Again, I suggest trying to visualize the whole board. If you do that, you will also see diagonals more clearly.