Thank you for sharing Gosia. The subject came up in a conversation with a friend this summer and I thought then: “I can compute the answer with the WL language”.
I would like to reply and expand on your post.
​
Now that you got me thinking about the subdivisions of the US states, I learned that most states in the US are subdivided into counties, except for Alaska and Louisiana which are called boroughs and parishes, resp.
In[]:=
states=EntityClass["AdministrativeDivision","AllUSStatesPlusDC"]["Entities"];
The counties, parishes and boroughs can also be calculated with the entity property “Subdivisions” :
In[]:=
Louisiana, United States
ADMINISTRATIVE DIVISION
["Subdivisions"][[1;;3]]
Out[]=

Acadia Parish, Louisiana, United States
,
Allen Parish, Louisiana, United States
,
Ascension Parish, Louisiana, United States

In[]:=
Alaska, United States
ADMINISTRATIVE DIVISION
["Subdivisions"][[1;;3]]
Out[]=

Aleutians East, Alaska, United States
,
Aleutians West, Alaska, United States
,
Anchorage, Alaska, United States

So the list of subdivision entities is simply this expression (which is slightly faster to calculate than mapping over an entity list of geoentities, and the result does not comprise of empty entities either).
In[]:=
subdivisions=
all US states with District of Columbia
ADMINISTRATIVE DIVISIONS
["Subdivisions"]//Flatten;
If you remove the subdivision designations i.e. county or parish, we can focus on the most popular names given to US states’ subdivisions. This can be achieved within the entity framework too:
In[]:=
names=EntityValue[EntityClass[subdivisions,All],EntityFunction[a,StringTrim[#,(" County"|" Parish")]&@*First@StringSplit[#,","]&@*CommonName@a]];
In[]:=
PopularSubdivisionNames=ReverseSort@Counts@names;
The subdivisions that appear in at least 25 states gives me just three names, not five:
In[]:=
Select[#>=25&]@PopularSubdivisionNames
Out[]=
Washington31,Jefferson26,Franklin25
In[]:=
PieChart[Select[#>=25&]@PopularSubdivisionNames,ChartLabels->Automatic]
Out[]=

Some subdivisions share the name of a state: how many are there?

We’ll use the common names of the states to select the subdivisions in question:
In[]:=
StateNames=EntityValue[states,EntityFunction[a,First@StringSplit[#,","]&@*CommonName@a]]
Out[]=
{Alabama,Alaska,Arizona,Arkansas,California,Colorado,Connecticut,Delaware,District of Columbia,Florida,Georgia,Hawaii,Idaho,Illinois,Indiana,Iowa,Kansas,Kentucky,Louisiana,Maine,Maryland,Massachusetts,Michigan,Minnesota,Mississippi,Missouri,Montana,Nebraska,Nevada,New Hampshire,New Jersey,New Mexico,New York,North Carolina,North Dakota,Ohio,Oklahoma,Oregon,Pennsylvania,Rhode Island,South Carolina,South Dakota,Tennessee,Texas,Utah,Vermont,Virginia,Washington,West Virginia,Wisconsin,Wyoming}
Now we can calculate how many subdivisions with the same names as states there are:
In[]:=
sameNames=Select[MemberQ[StateNames,#]&]@names//Counts//ReverseSort
Out[]=
Washington31,Delaware6,Wyoming3,Ohio3,Texas2,Iowa2,Nevada2,Mississippi2,Utah1,Colorado1,Indiana1,Oklahoma1,Oregon1,Idaho1,Hawaii1,District of Columbia1,Arkansas1

What subdivisions have the same name as the state they belong to?

​
First we will find all subdivision entities that share the name of a state:
In[]:=
stateNameEntities=FilteredEntityClass[EntityClass[subdivisions,All],EntityFunction[a,MemberQ[Keys@sameNames,StringTrim[#,(" County"|" Parish")]&@*First@StringSplit[#,","]&@*CommonName@a]]]["Entities"]
Out[]=

Washington County, Alabama, United States
,
Arkansas County, Arkansas, United States
,
Mississippi County, Arkansas, United States
,
Nevada County, Arkansas, United States
,
Washington County, Arkansas, United States
,
Nevada County, California, United States
,
Washington County, Colorado, United States
,
District of Columbia, District of Columbia, United States
,
Washington County, Florida, United States
,
Washington County, Georgia, United States
,
Hawaii County, Hawaii, United States
,
Idaho County, Idaho, United States
,
Washington County, Idaho, United States
,
Washington County, Illinois, United States
,
Delaware County, Indiana, United States
,
Ohio County, Indiana, United States
,
Washington County, Indiana, United States
,
Delaware County, Iowa, United States
,
Iowa County, Iowa, United States
,
Washington County, Iowa, United States
,
Washington County, Kansas, United States
,
Ohio County, Kentucky, United States
,
Washington County, Kentucky, United States
,
Washington Parish, Louisiana, United States
,
Washington County, Maine, United States
,
Washington County, Maryland, United States
,
Washington County, Minnesota, United States
,
Washington County, Mississippi, United States
,
Mississippi County, Missouri, United States
,
Oregon County, Missouri, United States
,
Texas County, Missouri, United States
,
Washington County, Missouri, United States
,
Washington County, Nebraska, United States
,
Delaware County, New York, United States
,
Washington County, New York, United States
,
Wyoming County, New York, United States
,
Washington County, North Carolina, United States
,
Delaware County, Ohio, United States
,
Washington County, Ohio, United States
,
Delaware County, Oklahoma, United States
,
Oklahoma County, Oklahoma, United States
,
Texas County, Oklahoma, United States
,
Washington County, Oklahoma, United States
,
Washington County, Oregon, United States
,
Delaware County, Pennsylvania, United States
,
Indiana County, Pennsylvania, United States
,
Washington County, Pennsylvania, United States
,
Wyoming County, Pennsylvania, United States
,
Washington County, Rhode Island, United States
,
Washington County, Tennessee, United States
,
Colorado County, Texas, United States
,
Washington County, Texas, United States
,
Utah County, Utah, United States
,
Washington County, Utah, United States
,
Washington County, Vermont, United States
,
Washington County, Virginia, United States
,
Ohio County, West Virginia, United States
,
Wyoming County, West Virginia, United States
,
Iowa County, Wisconsin, United States
,
Washington County, Wisconsin, United States

The subdivisions that have same name as that state they belong to, can now be filtered from the list of entities as so:
In[]:=
sameNameEntities=FilteredEntityClass[EntityClass[stateNameEntities,All],EntityFunction[a,SameQ[#[[1]],#[[2]]]&@StringTrim[#,(" County"|" Parish")]&@Part[#,1;;2]&@StringSplit[#,", "]&@CommonName@a]]["Entities"]
Out[]=

Arkansas County, Arkansas, United States
,
District of Columbia, District of Columbia, United States
,
Hawaii County, Hawaii, United States
,
Idaho County, Idaho, United States
,
Iowa County, Iowa, United States
,
Oklahoma County, Oklahoma, United States
,
Utah County, Utah, United States
