Several years ago I'v seen a book that describes facts that confirm that Russia (as well as USSR earlier) and USA are very close to each other, have a lot of similar in their evolution
Can you agree or not agree with that. And why

Sorry, but I cannot agree with that.

Russia is a country that developed slowly over centuries, a country that was born out of hardship, constantly under threat, suffering untold devastations, many invasions, a country almost constantly at war for 4 or 5 centuries.
Russia grew as an empire out of necessity, to repulse invasions, to contain threat and create distance with its enemies. Russia became a nation but including many different population, spreading the language and the culture.

The USA are quite different, they were originaly colonies created by British settlers against weak native opposition. The settlers had numbers, technology and a more advanced civilisation to impose their will and expropriate the land from the natives which they tried to eliminate.
Becoming independent after a quick war, the USA expended quickly through purchasing land from France, invading and annexing part of Mexico which could defend them.
The USA grew rapidely, attracting immigrants and offering land cheaply. The population increased dramatically during the 19th century, and took over the country from East to West, colonising everything without much opposition.

The history of Russia is one of long hard struggle to become a nation.

The history of the USA is one of constant superiority from the start.

That explains the different mentalities.