Real European gooseberries have been grown and used as food for centuries. Maybe for this reason other fruits have also been given the name gooseberry even though none of them really bare any resemblance to the real European Gooseberry. It does cause much confusion especially with Indian Gooseberries.
Real European gooseberries grow on thorny shrubs which are cold and somewhat shade tolerant. They grow well in the Northern US not so well in southern states where it tends to be too hot.
Indian Gooseberries are much larger fruit that grows on tropical trees. They are used extensively in Ayurvedic medicine and are also tart leaning more towards bitter. They cannot grow in the US its too cold. Perhaps in southern Florida.
Cape Gooseberries often called golden berries are related to the tomato. These grow on soft vigorous bush like shrubs. They can be grown from zone 6 - 10. In zones 6 and 7 they are grown as annuals but will act as perennials from zone 8 onwards. The bushes are soft and tender and cannot tolerate any cold.
Chinese Gooseberries have now been renamed as kiwi fruit mostly as a marketing ploy. These grow on vigorous vines. There are two kinds the large ones that can be purchase in most grocery stores are semi tropical and hardy kiwi berries that are much smaller, and have a lot more flavor. These can be grown to zone 6 but are very susceptible to late frosts.