The major similarity between these two is in their ideas about inequality.
Alexander Hamilton believed that society should be somewhat hierarchical. He, unlike his rival Thomas Jefferson, did not believe that everyone in the society should be equal in economic or even in political terms. Romney would not go this far, but he is much more accepting of economic inequality than President Obama is. Romney and other conservatives believe that economic inequality is good for a society because it gives the working and middle classes an incentive to work harder so they, too, can get the material wealth they see being enjoyed by others above them on the ladder. Conservatives believe that a more equal distribution of wealth would blunt the motivation that pushes people to work hard.
We should also note that both men are similar in terms of their desire to have government help business. Early in our history, Hamilton pushed for government action to make a better climate for business so that businesses could thrive. Romney, too, is inclined to implement policies that allow businesses to grow more easily even if those policies might (in some people’s minds) do things like increasing inequality or hurting the environment.