In the UK and Ireland, as far as I can tell, county names either start with "county" (e.g. County Down), end with "shire" (e.g. Shropshire), or have a name that includes no recognizable morpheme meaning "county" (e.g. Essex). In the US, on the other hand, counties tend to have the name "county" at the end (e.g. Orange County). ETA: This is even true when the US county in question is named after a British or Irish county, e.g. Essex County in New Jersey.
How did this discrepancy come about?
Counties in the United States today are totally different entities than counties in the British Isles (by which, reader forgive me, I include Ireland, for historical reasons that shall become clear).
In Great Britain and Ireland, “counties” are today largely ceremonial or historical entities, whose borders are the results of centuries of historical happenstance. Counties today in those countries only occasionally coincide with contemporary administrative or political divisions or sub-division.
In the United States, by contrast, counties get created de novo by state (and earlier colonial) governments as administrative units often in response to local political agitation for political centers to be more easily accessible to local land-owners. This system of counties as the initial and principal sub-unit of state governments originally is in the 18th century loosely modeled on the English and Welsh county governments at the time, but as early as the 1780s the functions and conceptions of a “county” had begun to diverge significantly.
American counties are the principal local administrative below the state level, and administer key government functions such as local civil courts, tracking land ownership, administration of property taxes, maintaining much of the local road network, and providing primary or supplemental policing services. In rural areas, outside of incorporated cities or towns, the county government is the provider of key government services, including road repair, law enforcement, and even primary education.
American counties serve a different administrative and social purpose than British or Irish counties do today, and the resemblance in naming is more ancestral than due to underlying similarity.