Is this safe?
You should have a certain level of trust on the website you decide to connect your wallet with, and that they ensure there is no malicious third-party scripts which would intend to read the wallet connection configuration, either from memory or storage. We heavily recommend to add CSP rules to your site and follow best practices to prevent XSS.
Connectors with budget controls or confirmation dialogs (Alby extension or NWC) are recommended so you have full control over your connection.