This is an operating system agnostic question. We are recommended to use password managers which encourage us to copy and paste our passwords. What protections exist if any to say a malicious website in another tab copying clipboard content?
What protections exist if any to say a malicious website in another tab copying clipboard content?
This is handled at the web browser level. There is no API by which a webpage can read the clipboard without explicit user interaction *. It would only be possible with a browser exploit, in which case you would probably have bigger problems.
* Obsolete IE versions had an option where this could be enabled, but it was not enabled by default.
There are a couple assumptions being made in your question. First, the OS does not protect the clipboard. Text there is in memory until overwritten. It's not meant to be a sensitive process that requires protection from different processes under the same user context.
Secondly, even though your question is meant to be OS agnostic, by necessity, it is required to be aware of how various password managers work. Some managers copy to the clipboard then overwrite the clipboard entry after a few seconds. Some do not use the clipboard, and some use alternate methods of automatically entering the password into fields.
If you mean that the user copy/pastes from a source (password manager, text file, etc.) then the user needs to manage this problem themselves.
Third, if there is malware running in "another tab" that can read user memory or the memory space of another tab, then there are much bigger problems to worry about than protecting clipboard memory.