Well, wbogacz already gave a very good answer to the question about the chronology and its clarity. But as to if it was intended or how far it is consistent with the overall story, I think it is rather us audience who just want to draw a connection between different characters where there is none. This motivation lead at least to the coining of the term "Man With no Name"-trilogy, although this man is a completely different character in the particular movies. You wouldn't believe Sentenza to be the same person as Colonel Mortimer just because they were portraied by the same actor, would you? So there isn't anything that makes me believe Clint Eastwood's character to be the same person in The Good, The Bad and The Ugly as in A Fistful of Dollars.
This is to some degree supported by the corresponding Wikipedia article which states:
Although it was not Leone's intention, the three movies came to be considered a trilogy following the exploits of the same so-called "Man with No Name" (portrayed by Clint Eastwood, wearing the same clothes and acting with the same mannerisms). The "Man with No Name" concept was invented by the American distributor United Artists, looking for a strong angle to sell the movies as a trilogy. Eastwood's character does indeed have a name - albeit a nickname - and a different one in each film: "Joe," "Manco," and "Blondie," respectively.
So given that the connection of the third movie to the others was developed after the fact, there is likely no intentional connection between the characters and story of those movies, even if the chronology is pretty definite. Instead of by their backstory and identity, the characters played by Clint Eastwood in those films are rather connected by their traits, mannerisms and style, which might suggested a deeper connection where there is none. And in fact this is IMHO by-design. We are just not supposed to know anything about this man's motives and his story before or after the individual movies, as that's what actually makes him
the a "Man with No Name".