There is at least one U.S. state in which the political party that is affiliated with the Democratic party goes by another name.
In Minnesota, they have the Democratic-Farmer-Labor party (DFL for short) rather than the Democratic party at the state level.
In New York State, they have the plain old "Democratic Party" and "Republican Party", but candidates can run as candidates for multiple political parties at the same time. So, in New York State, for example, James Monroe could run as a candidate for U.S. Senate on the Democratic Party ticket, the Communist Party ticket, and the Libertarian Party ticket, at the same time in the same election.
The relationship between elected officials once they are elected, and political parties, is mostly a voluntary one based upon mutual admiration and loyalty. National parties are usually built on the foundation of many local political parties that mostly cooperate every four years to try to elect a President from the party.
State parties typically have formal organizational control of their local branches within the state, but from a practical perspective, they have no real capacity to force a local elected official to comply with their directives, rather than defecting and becoming an independent elected official who is no longer a member of any political party, and in many localities, most partisan elected officials who are incumbents could successfully pull that off if they were inclined to do so.
At the local government level, there are some local governments that have traditional partisan elections (especially county governments).
In the case of local governments with partisan elections in the U.S. those political parties are almost always integrated into the state political party structure. But, in the U.S. there isn't all that much coordination at a policy level between local government elected officials, state government elected officials, and federal government elected officials. In contract, in most European countries, there is quite intense political coordination between elected official of the same party at different levels of government.
There are also a great many local governments (especially municipalities, New England towns, school boards, and "special districts") that have elections that are formally organized on a non-partisan basis. But, in these "non-partisan" local governments, it isn't all that unusual for informal slates of candidates or one of a kind local political parties to emerge.
For example, the small city of Glendale, Colorado (an urban enclave with about 5,000 residents), has had a political movement of candidates for city office running under the banner of the Glendale Tea Party (a name chosen in jest before the "Tea Party" had become the name of a very conservative faction of the national Republican party in the United States).
Similarly, in many school districts in Colorado, it isn't uncommon for the local teachers union to endorse one slate of candidates. In other districts informal civic groups will organize a slate of "reform" candidates for school board, and in at least one exurban Colorado school district, a bunch of people who were also leaders in the county Republican party where the school district was located put together a slate of unofficially Republican school board candidates to square off against a teacher's union slate of candidates.
These informal local political parties often have no formal legal existence, exist in only one or two different local governments at any one time, tend to be focused on political issues particular to the locality, and tend to last for only one, or at most, just a few, election cycles. While these informal groups often coordinate the process of getting campaign donors to give to multiple people on the same slate, these informal political parties rarely have meaningful coffers of their own to fund campaign.