Persian is English word used for Farsi language which is spoken in today’s Iran, Afghanistan, Tajikistan, Uzbekistan and parts of Turkmenistan.
The same way when we have British English, American English, Australian English and Canadian English or the same way we have Egypt Arabic, Saudi Arabia Arabic, Libya Arabic so we do have Afghanistan Persian.
Persian is a united language but the due to spread of the language in large geographical area it creates different accents and flavors. I’ll give you an example. In Britain when people stand on the line to buy a ticket or something, they call it “queue” so they say “stand on the queue”. But in America they say “stand on the line” but Americans know the meaning of “queue” and British know the meaning of “line” they just used different terms for their purpose.
The same way people of Afghanistan and Iran can read each others writing and they use common language but geographically they use different terms for different things.