In your defence, they are when you open up a German dictionary. Bayern is the name of the state of Bavaria, while Bayer is the word for a Bavarian person. So, in the case of Bayern München, the name means Bavaria Munich.
It's not an uncommon way of naming teams in Germany. Think about the two Borussias from Dortmund and Mönchengladbach. Borussia is the Latin name for the former kingdom and German state of Prussia that stretched across much of northern Germany, including the area where Dortmund and Gladbach are located. So, just think of it as Prussia Dortmund or Prussia Mönchengladbach.
The cities of Dortmund and Mönchengladbach are close and in the same state, but it's not a derby when the two meet, despite what the name may suggest. That is also a common misconception when Bayern and Bayer face off. The names would suggest they're close, but Munich and Leverkusen are almost 300 miles apart.
When looking to discern location in a German team name, your best bet is the last word. For example, Borussia Dortmund, Werder Bremen, Eintracht Frankfurt, Hertha Berlin etc.
That's different from naming conventions in English-speaking cultures, where you mostly have the location at the start, followed by an array of suffixes like United, City, Town, County and more.
We can now hear you asking: "So what about Bayer Leverkusen if they're not Bavarian Leverkusen?"
In their case, it's quite simply a quirk of fate that means it looks similar. Die Werkself were created in 1904 by workers for local pharmaceutical company Bayer as part of sports club TSV Bayer 04 Leverkusen. The Bayer company takes its name from founder Friedrich Bayer.
Bayer (with variants including Baier, Beyer and Beier) can be derived from the Bayer that means Bavarian, but it can also have an occupational derivation from beiern (to ring a bell), referring to people who would ring church bells.
So, there is a potential link between the two club names, but the origin of the Bayer family name is lost to time. Maybe without realising he has given his name to a club called Bavarian 04 Leverkusen.
But in reality, there's no connection between the Bayern of Munich and the Bayer of Leverkusen. The former is a location, the latter a family name. They just happen to look similar. So, make sure you watch carefully for your Bayerns and Bayers.