I bought the app on Android, do I have to buy it on iOS too?

Short answer: Yes

Long Answer:

Since purchases are made through either Google (Android) or Apple (iOS — iPhone, iPad), and these companies don't exchange app licenses with each other, the app has to be purchased on each platform separately.

Also, developers don't see who bought their apps, so it isn't possible for me to verify that a purchase was made by a given person and, for example, generate a promo code to give the app for free to someone who bought it on the other platform.

Finally, I had to create the app from scratch on each platform. The apps access low-level network APIs of the operating systems, which differ considerably, and I couldn't reuse any code. So even though I tried to make the apps similar on both platforms, they are in fact completely different code-wise, and I find it fair to get paid for the time I spent developing them.