Most of the crypto wallets do not require people to complete the KYC verification. Though mostly the non-custodial wallets are the only ones on that list, as custodial wallets do require KYC verification.
Trust Wallet does not require its customers to go through this verification to access the basic functionality offered by Wallet. The only goal of this wallet is to provide users with a decentralized and non-custodial solution for storing and the management of cryptocurrencies without the need for excess verification processes. When you download and install this wallet on your mobile device or laptop you can immediately start using it to securely store various cryptocurrencies, such as Bitcoin, Ethereum, and many others, without having to verify your identity. If you stumble across any website or page, urging you to enter the personal details to complete the verification then get a hint that it is a fraudulent page. Thus, try to leave the page you are on as soon as possible and ignore any message or email you receive in the name of Trust Wallet KYC verification. However, it's important to note that certain features within Trust Wallet may require additional verification or authentication. You will get to learn those features when you start to operate the wallet.