Honestly all the factions got some dose of America in them from Americans descendants moving from the inner colonies to the outer or what would become the Great Houses. There are American named planets across the Inner Sphere. The Davion family came from North America. Well specifically Canada is where the Davion family came from before Scotland and Avon, France. No one is really Space America, but the closest is the Rim Collection who have an elected president for a certain term and legislature.
The Combine got several worlds that are really American like Midway. The Capellan honestly got the same deal with worlds like Old Kentucky. However, the Free Worlds League and Federated Suns got the lionshare of the legacy other than the Terran Hegemony. The Lyran Commonwealth got a larger portion of the legacy as well, but there is something different about their attitude. Then again, I always got a Hanoverian England Period attitude from them.
But the two big inheritors were the Free Worlds League and Federated Suns. Free Worlds League got the whole legislature and democratic principles. The Suns marginalized their legislature. Their inheritance is something different; plus it can easily come from England as well though it is with an American twist. Though the League has a constitution in its Document of Incorporation, I'm not sure what is in it, but one odd thing is in there that allows Parliament the ability to readjust rights. However, that does not seem to include Lockean Rights. In fact, I'd say that the highest power in the Free Worlds League is Parliament when it comes to what are the individual citizens civil rights which can be altered. Which is honestly closer to British Parliaments. The Home Defense Act is closer to the pre Cold War America raising state militias rather than a true standing army in times of need.
Now before the Lyran fans go crazy, I know the Document of Acceptance has some form of rights for the citizens, but it is vague and not really well defined either. The Suns to my knowledge is the only state that has a Lockean list of civil rights aka the Six Liberties spelled out in the Crucis Pact. It's the closest to any of the governments in Battletech to the American Bill of the Rights since it defines what the FedSuns government can't do to its citizens. And in fact in addition to speaking English is probably the leading reasons why a lot people make the assumption that it is Space America. Other than the argument that since it was the antagonist fighting against the Cold War evil stand-ins that it became Space America.
The truth is the Suns was founded on similar lines as the USA, but due to alterations, it became Space Camelot. It isn't Space France either. The problem with calling the Suns Space France is that it is not Republican France of Liberté, Égalité, Fraternité. And it is not the France of Louis the XIV of "Je suis l'etat." That spirit is honestly embedded in the Capellan Confederation in the idea of the Chancellor being the only interpreter of Greater Humanity, and thus the position of Chancellor becomes the embodiment of the Capellan state.
That's not true with the Suns. Davion First Princes are bound by the law in the Suns, and it was why Katherine, Edward the Terrible, and the Varneys faced such problems in the Suns when they tried to change things. All tried to take away the rights of the people of the Suns, and each of those groups were defeated by popular movements eventually lead by another member of the Davion family. Then again, the populace is armed as well, so it's hard to be an absolute dictator in the Suns if you break the Six Liberties.
However, what makes the Suns different from America is the adoption of nobility and the President of the Suns becoming the First Prince. Which is something that happen to all of the protopowers mostly due to probably to the sheer amount of time that it took to administer a national election. Plus, the Suns is highly clothed in the ideas of well Camelot. (That is also why I have a distaste for the novel, the Ideal War. It mostly ignores that the Suns is basically Space Camelot that many names are taken from Arthurian Legend and that it was highly detailed in the original Housebook and the Warrior Trilogy. Plus eventually due to the Free Worlds League lack of knowledge about Arthur which is a plot point in the Ideal War, that faction gets Camelot named warships. UGH! Oh well, Yvonne got them back when she named that captured Blakest vessel, the Excalibur. Oh and the Knights of the Inner Sphere got killed. Yeah!)
But the Steiner did get one thing of America, they got Dixie and vague form of checks and balances that doesn't often work that well i.e. Alesandro Steiner and Katherine Steiner-Davion.
So in conclusion, the Free Worlds League got the antebellum USA's state issues. The Steiners got Dixie and probably southern cuisine to go with it. The Draconis Combine got planet names and probably descendants of the USA though everyone else did as well. The Capellan Confederation got moonshine from Old Kentucky. And the FedSuns got Lockean Rights.
And the Clans got a rock from the Terran Hegemony that spelled out how to be Space America as well. Nicholas promptly ground it up and went in a totally different direction.