Uniswap v3 was launched under a Business Source License 1.1 (BUSL 1.1).
The license limits use of the v3 source code in a commercial or production setting until April 1st, 2023. After this, the license will convert to a general public license. This means anyone can fork the code for their own use — as long as it is kept open source.
The Uniswap Labs team has created documents that outline the forking process here.
Updated