Falls Church City, VA is not in any county, politically. Although, it is surrounded by Fairfax County and Arlington County. Its borders are based on political decisions made in the late 19th Century. At the time of its formation, Fairfax County included all of what are now Loudoun and Arlington counties, and the cities of Alexandria, Falls Church, and Fairfax.
The Commonwealth of Virginia has unusual political geography. It contains independent cities and towns that are not part of any county, politically, despite in many cases being entirely embedded in that county. Falls Church City, Fairfax City, and the City of Alexandria are all examples of this phenomenon. There are 38 such independent cities in Virginia, and three in the rest of the US.
Confusingly, the US Post Office recognizes parts of Fairfax County as "Alexandria" and "Falls Church" but these are informal and do not mean anything as far as taxes and government. "Falls Church High School," for example, is part of the Fairfax County school system and has no relationship to FC city or to George Mason High School, which serves Falls Church City.
Because the US Census reports Virginia independent cities separately from the surrounding county, this can lead to confusing results due to very small reporting unit sizes - Falls Church City has one of the highest per capita reported incomes of any "county" in the US, for example, but has a total.population of only 13,000, and is entirely urban.
Falls Church city lies to the west of Arlington County and Fairfax County surrounds the rest of Fairfax City. Residences in Fairfax County near Falls Church city have Falls Church mailing addresses. Mailing addresses are tied to the post office that delivers your mail, not where you live. Arlington County was originally part of Washington DC. That defined part of Falls Church city’s boundary. Thus, the City of Falls Church is an independent jurisdiction, outside of nearby Arlington and Fairfax Counties and not part of the City of Alexandria. However, there are a number of addresses, for example in Bailey’s Crossroads and at Skyline which has a Falls Church postal address, and yet are not in the City of Falls Church.