Not exactly the same name, but American flies from Charlotte Douglas International (CLT) in Charlotte, North Carolina to Cyril E. King Airport (STT) in Charlotte Amelie, U.S. Virgin Islands, though the latter is usually advertised as Saint Thomas, which is the island where Charlotte Amelie is located.

The US low-cost airline Avelo is launching flights between Wilmington, Delaware (ILG) and Wilmington, North Carolina (ILM).

First I’ll propose one historic but concrete example there used to be a helicopter shuttle between London (LHR) and London (LGW) (same city, different airports). I’m sure there were similar services in other countries.

Now for my attempts to find something current. I don’t believe there is anything, but can’t claim to have conclusive proof.

From Wikipedia’s list of international airports I extracted all the airport names (pasted into a text editor, tidied up, piped through sort. Of course at this point I may already be omitting some domestic airports like Portland(PWM)-Portland(PDX) as mentioned in the comments.
The full list is too long to include here but is on Dropbox.

Then I used a Python script to pare this down to duplicate city names, generate URLs for Google flights, and load those URLS (rate-limited) in the default browser:

    #data taken from https://en.wikipedia.org/wiki/List_of_international_airports_by_country
import codecs
import webbrowser
import time
airport_list=[]
duplicates=[]
testURLs=[]
base="https://www.google.com/flights?hl=en#flt="
date1="2020-11-28"
end=".2020-12-02;c:GBP;e:1;sd:1;t:f"
with codecs.open('airports_sorted.txt', encoding='utf-8') as airports:
    for line in airports:
        airport_list.append(line.replace('\n','').split("\t"))
        
print ('%d total airports found'%len(airport_list))
for n,airport in enumerate(airport_list):
    for m,possible_duplicate in enumerate(airport_list):
        if airport[0]==possible_duplicate[0] and n!=m and airport[1] not in ([duplicate[1] for duplicate in duplicates]):#test for same city name, not same index, not same airport name
            duplicates.append(airport)
            testURL=base+airport[2]+"."+possible_duplicate[2]+"."+date1+"*"+airport[2]+"."+possible_duplicate[2]+end
            testURLs.append(testURL)
print ('%d airports with duplicate city names found'%len(duplicates))
print ('They are:')
print ('City\tAirport\tCode')
for duplicate in duplicates:
    print ('%s\t%s\t%s'%(duplicate[0],duplicate[1],duplicate[2]))

tab_count=0
for testURL in testURLs:
    print (testURL)
    webbrowser.open_new_tab(testURL)
    tab_count+=1
    time.sleep(1)#load slowly
    if tab_count>9:
        dummy=input("press enter to continue")
        tab_count=0#load max 10 tabs at once

Leaving the following list of 70 airports:

Alexandria  Alexandria International Airport    ALY
Alexandria  Borg El Arab Airport    HBE
Bandung Husein Sastranegara International Airport   BDO
Bandung Kertajati International Airport KJT
Beijing Beijing Capital International Airport   PEK
Beijing Beijing Daxing International Airport    PKX
Birmingham  Birmingham International Airport    BHX
Birmingham  Birmingham-Shuttlesworth International Airport  BHM
Buenos Aires    Aeroparque Jorge Newbery    AEP
Buenos Aires    Ministro Pistarini International Airport    EZE
Chicago Chicago Midway International Airport    MDW
Chicago O'Hare International Airport    ORD
Colombo Bandaranaike International Airport  CMB
Colombo Ratmalana International Airport RML
Columbus (OH)   John Glenn Columbus International Airport   CMH
Columbus (OH)   Rickenbacker International Airport  LCK
Frankfurt   Frankfurt Airport   FRA
Frankfurt   Frankfurt-Hahn Airport  HHN
Houston George Bush Intercontinental Airport    IAH
Houston William P. Hobby Airport    HOU
Istanbul    Istanbul Airport    IST
Istanbul    Sabiha Gökçen International Airport SAW
Jakarta Halim Perdanakusuma International Airport   HLP
Jakarta Soekarno Hatta International Airport    CGK
Johannesburg    Lanseria International Airport  HLA
Johannesburg    OR Tambo International Airport  JNB
Kristiansand    Kristiansand Airport, Kjevik    KRS
Kristiansand    Kristiansund Airport, Kvernberget   KSU
Milan   Linate Airport  LIN
Milan   Malpensa Airport    MXP
Moscow  Domodedovo International Airport    DME
Moscow  Sheremetyevo International Airport  SVO
Moscow  Vnukovo Airport VKO
New York City   John F. Kennedy International Airport   JFK
New York City   LaGuardia Airport   LGA
Oslo    Oslo Airport-Gardermoen OSL
Oslo    Sandefjord Airport  TRF
Panama City Northwest Florida Beaches International Airport ECP
Panama City Tocumen International Airport   PTY
Paris   Charles de Gaulle Airport   CDG
Paris   Orly Airport    ORY
Reykjavik   Keflavik International Airport  KEF
Reykjavik   Reykjavik Airport   RKV
Rio de Janeiro  Rio de Janeiro–Galeão International Airport GIG
Rio de Janeiro  Santos Dumont Airport   SDU
São Paulo   São Paulo–Congonhas Airport CGH
São Paulo   São Paulo–Guarulhos International Airport   GRU
Seattle Kenmore Air Harbor Seaplane Base    LKE
Seattle King County International Airport   BFI
Seattle Seattle-Tacoma International Airport    SEA
Shanghai    Shanghai Hongqiao International Airport SHA
Shanghai    Shanghai Pudong International Airport   PVG
Singapore   Seletar Airport XSP
Singapore   Singapore Changi Airport    SIN
Stockholm   Stockholm-Arlanda Airport   ARN
Stockholm   Stockholm-Bromma Airport    BMA
Stockholm   Stockholm-Skavsta Airport   NYO
Stockholm   Stockholm-Västerås Airport  VST
Tehran  Imam Khomeini International Airport IKA
Tehran  Mehrabad International Airport  THR
Tenerife    Tenerife North Airport  TFN
Tenerife    Tenerife South Airport  TFS
Valencia    Arturo Michelena International Airport  VLN
Valencia    Valencia Airport    VLC
Victoria    Seychelles International Airport    SEZ
Victoria    Victoria International Airport  YYJ
Warsaw  Frédéric Chopin Airport WAW
Warsaw  Modlin Airport  WMI
Washington, D.C.    Ronald Reagan Washington National Airport** DCA
Washington, D.C.    Washington Dulles International Airport IAD

None of these pairs show anything on FlightRadar24’s search of direct flights, but that only searches current flights. Google similarly doesn’t find anything direct. The best is Panama City (Panama, PTY) to Panama City (Florida, ECP) with a change in Houston. Google should hopefully find everything direct, but it certainly misses some indirect routes: flightconnections.com finds Valencia (Spain, VLC) to Valencia (Venezuela, VLN) with 2 changes (several routes) but Google doesn’t.

I won’t claim this is conclusive for a couple of reasons:

• As I noted above it misses domestic airports (if there’s a list of all airports in the world, I can try pointing my script at that).
• Some airports may serve more than one city, or not the city it’s named for (e.g. Newark is generally thought of as serving New York). That would be hard to code, but harder to define.
• This depends on the completeness and correctness of the list, which I can’t vouch for. I found one typo: KSU serves Kristiansund not Kristiansand, so there’s a fake duplicate on my list; a typo with the opposite effect could hide a success. Also ALY isn’t an airport serving Alexandria. It might have been in the past, or it might have been called AEX.

There is now a direct train 041/042 from [Veliky] Novgorod to [Nizhny] Novgorod and back.

And nobody is very happy about it since it messed up former city residents’ arrival time to Moscow.

Since your rules do not mention that flights have to be regular nor commercial, some “test flights” may satisfy yor criteria.

The ticket below was for a direct flight from Toulouse (TLS) to the very same Toulouse (TLS) airport, approx. 10 hours long just to reach the place you were departing from.

This was the first Airbus A340 flight with passengers, aimed at testing the cabin and its related facilities (intense use of toilets, heating and serving hundreds of meals, etc.). Except that you could not buy the ticket from an airline, it was just like a normal, fully booked, evening flight. You would pop up at the airport 1-2 hrs before, go through security, wait in the terminal and board the plane from a gate. During the flight, the plane just circled above the Pyrénées mountains to make time pass and burn its fuel. It then returned to its original airport in the morning (no jetlag!).

Funny start?

Springfield SFY (MA) to Springfield SPI (IL) doesn’t result in any connection for my booking advisor, neither to Springfield SGF (MO) nor Springfield VSF (VT). Neither does any other combination yield results, nor does Marshall MLL (AK) to the Marshall Islands MAJ nor St. Petersburg PIE (FL) to St. Petersburg/Leningrad LED (Russia)^{too bad, we could have listened to the song of the same name on the flight}, nor could any combination of Newcastle/New Castle to New Castle be found.

4

To get from Alexandria AEX (LA, USA) to Alexandria HBE (Egypt) it takes 3 to 4 stops, so sadly that’s out. Most ancient Alexandrias lack airports.

3

Glasgow GLA (Scotland) to Glasgow GCW (MT) has 3 stops. As is Trinidad (TTD) to Trinidad-Tobago, a country, using Port of Spain POS as a mark. Or Armenia AXM in Colombia to Yereven EVN in Armenia.

2

From Lebanon LEB (NH), it’s 2 stops to Beirut BEI in Lebanon, so… kinda on/off the mark again? But remember Newcastle? Newcastle NCL (England) connects with Newcastle NTL (Australia) via 2 stops. Melbournde MEL (Australia) to Melbourne MLB (FL) has, once again, 2 stops. Roma RMA (Australia) to Roma ROM (Italy) usually has 2 or 3 stops.

More 2 stop: La Paz LPB (Bolivia) to La Paz LAP (Mexico). Sydney YQY (NS) to Sydney SYD (Australia). Augusta AGS (GA) to Augusta AUG (ME). Aberdeen ABZ (Scotland) to Aberdeen ABR (SD). Kandahar KDH (Afganistan) to Alexandria HBE (Egypt), because Kandahar is one of the 20-something Alexandrias. Or Kandahar KDH to Khujand LBD (Tajikistan), also founded as… Alexandria. Or from Khujand LBD to Termez TMJ (Uzbekistan)…

1

Kingston KIN (Jamaica) to Kingston YGK (ON, Canada) is 1 or 2 stops, we get better!
Same for London LON (England) to London YXU (ON), 1 stop and occasional 2 stops, as is San Jose SJC (CA) to San Jose SJO (Costa Rica), as is from San Jose SJO to San Jose SJD (Mexico), or Santiago SCL (Chile) to Santiago STI (Dominican Republic), or Manchester NHT (NH) to Manchester MAN (England) or Rochester ROC (NY) to Rochester RST (MN). Or Manhattan MHK (KS) to Manhattan, New York NYC (NY), landing in EWR or JFK.

But to get from San Jose SJC (CA) to San Jose SJD (Mexico) or Columbus CMH to Columbus CSG (GA), it’s always 1 stop, Again, a little better on the track. Same for Washington (DC) WAS to Washington (State) using SEA. Or Saint John YSJ (Canada) to St.John’s YYT (Canada). And another case of former-Alexandrias: Kandahar KDH to Herat HEA (both Afganistan) via Kabul.

my winner is:

• Santa Cruz (de la Palma) SPC Santa Cruz (de la Tenerife) TCI (both Spain) – which is a 30 mintues nonstop domestic flight in Spain.

Honorable mention

Kaliningrad (since 1996: Korolyov) to Kaliningrad KGD (formerly: Königsberg) in Kaliningrad Oblast would have been a nonstop flight… from Moscow SVO.

The mad Alexandria-dash, always taking 2 stops inbetween ancient Alexandrias:
Alexandria HBE – Kandahar KDH – Khujad LBD – Termez TMJ

Pfäffikon SZ and Pfäffikon ZH are two suburbs of Zürich, about 10km from each other. SZ (Schwyz) and ZH (Zürich) refer the the cantons these villages are in.
Both are on the Zürich suburban train network, and you can travel between them every half hour. At one time there were even direct trains…

Newark, NJ to Newark, DE by Amtrak train.

5:15 PMNewark Penn Station
Acela Express 2167 towards Washington Union Station
6:27 PM
6:40 PMWilmington Station
Wilmington/Newark Line 3257 towards Newark
6:59 PMNewark
(Newark, Delaware)

Direct flight from EWR (Newark, NJ) to ILG Greater Wilmington Area (7 miles from Newark, DE)
https://www.flights.com/flights/wilmington-ilg-to-newark-ewr/

Alaska Airlines flies nonstop from San Jose, CA USA to San Jose del Cabo in Mexico.