In both cases, the copies will be placed in 'My Drive', which you should move to a sub-folder.

If its parent is a shared folder, select the files you want to copy, right click on the selected files and select 'Make a copy'.

If its parent is 'Shared with me', right-click on the file and select 'Make a copy'. To ensure that you keep someone else's files permanently, make a copy. It is always possible for the other person to permanently delete their files (often this would simply arise from technical reasons, such as deleting their account, reducing storage, or accidents, rather than an unkind decision). In addition, it prevents people from believing that other people's files are permanent. This does simplify syncing, as far less data is used to sync a shortcut, compared to large items. However, items can no longer be placed in more than one folder. This made syncing other people's files with one's computer's hard drive easy. Previously, one was able to add files and folders from 'Shared with me' to one's Drive. This is a recent change (from around April 2020).

Shared with me google drive app