Bananas are berries; strawberries are not
Botany's 'berry' and the supermarket's are two different things.

1In botany, a berry has a strict definition: a fruit developed from a single ovary, with seeds embedded in the flesh. By that rule, bananas, kiwis, watermelons, aubergines and tomatoes are all berries — but strawberries are not: the 'seeds' on the outside are the true fruits (achenes), and the red flesh is just enlarged flower tissue.
2Everyday names and scientific categories mismatch everywhere: raspberries and strawberries carry 'berry' in the name yet are not botanical berries; avocados are; almonds, despite being called nuts, are drupes — stone fruits, kin to the peach. To classify a fruit you must ask which part of the flower it grew from, not what it is called.
3One more twist: bananas are rich in potassium, which contains the naturally radioactive isotope potassium-40, so bananas are very slightly radioactive — scientists even use the playful 'banana equivalent dose' to explain radiation. No need to worry: you would have to eat millions at once to approach any danger.
Botanically, a berry comes from a single ovary: bananas qualify, strawberries do not.