How did Henry Clay's "American System" contribute to the creation of a uniquely American culture between 1800 and 1825?
It is a little bit difficult to argue that the American System helped lead to a "unique" American culture different from that of any other industrialized country. Much of what Clay's system did was to make America move towards industrialization like many other countries. This was particularly true of the national bank and the tariffs.
What this system did to, to some extent, was to contribute to making the United States more of a nation. That is, the American system helped to tie various parts of the country together. This is particularly true of the internal improvments to transportation. These improvements helped to create a national market where people from one area of the country could sell to those in another area. This helped to make the various parts of the country feel that they had more in common with one another.