The 3 Republican presidents of the 1920s were lumped together as identical. Is it right to say this or were there personal or political differences between them?
The three Republican presidents of the 1920s were quite different. Harding was a womanizer who enjoyed alcohol. He appointed his friends to key government positions; these people ultimately robbed the country and tarnished his reputation.
By most accounts, Harding is viewed as a failure. Coolidge, his successor, was morally quite upright. Coolidge did not interfere in business because he did not view it as the place of the federal government to intervene in economics.
Hoover is quite a shift from the other two presidents. Hoover, Coolidge's secretary of commerce, was viewed unfavorably by Coolidge, who thought he sought too much power. Hoover took actions during the Great Depression such as attempting to get businesses to maintain wages and using loans to bail out homeowners. Hoover also authorized public works projects such as Hoover Dam.
While Hoover was considered a failure by those who lived through the Great Depression, later historians give him more credit. Some of his programs were stalled by a Democratic Congress who wanted to see him lose in 1932. While Hoover took some action to limit the power of the Depression, the moves were not radical enough and came too late to save the economy. While Hoover's presidency cannot be considered a great success, he was not a failure like Harding and he was significantly more dynamic than Coolidge.
I am assuming that you are asking about the three Republican presidents of the 1920s. I have edited your question accordingly.
There were certainly major similarities between these presidents. The most prominent of these was that all of them were generally pro-business. They were all opposed to the sorts of reforms that Progressivism had pushed.
However, they were very different, especially in personal terms. Harding, for example was something of a playboy. By contrast, Coolidge was known as "Silent Cal" and was a serious family man (though he did enjoy dressing up as an Indian, of all things, and having his picture taken).
Politically, Coolidge was the most laissez-faire of the three. Hoover, by contrast, felt that government and business should cooperate. Where Coolidge thought that government should stay out of the way of business, Hoover thought the two should work together to improve the nation.
These three presidents had common attitudes, but they were certainly not simple copies of one another.