When adding the Lace extension to Chrome you will notice a pop-up stating that Lace will read your browsing history. Declaring requests such as this to users is a requirement from Google for anyone publishing to the Chrome store (more on that here).
When a DApp requests a connection to Lace using the CIP-30 - Cardano DApp-Wallet Web Bridge (the DApp connector), Lace uses the Chrome Tabs API to pull information from the DApp to read its URL, name, and icon so that this information can be displayed inside Lace. By using the Tabs API, Lace is explicitly required to declare these permissions – just as any other extension employing CIP-30 would have to do.
Rest assured, no other information is read or retained by Lace. The Lace team and IOG are absolutely committed to respecting user privacy and will not use this data in any intrusive or inappropriate ways.