While I do not use 1Password to manage my ssh keys, it sounds like you do and that you are being prompted to authenticate (or "approve") with 1Password in order to use a private key for each connection. This strongly suggests that you are not using the 1Password ssh agent and/or do not have ssh configured to use it.
When an ssh agent is working, you should only need to authenticate once for your unencrypted private key to be provided to the agent, which then provides it when needed to make connections, however the default for 1Password's agent is to require new "approval" when the 1Password client times out and locks and in other conditions. See Step 6: Authorize the SSH request.
If you believe that you have the agent working and ssh configured properly but are still having problems, then I recommend showing - not just telling - what happens when you run ssh to make a connection along with relevant commands and output showing details of the agent and ssh configuration.
A pedantic point to assist with clarity: your problem is substantially unrelated to a kernel (Linux), your shell (bash), your unspecified terminal emulator, the multiplexer (tmux), or logins - all of which you mention, but with ssh - which you didn't mention. When testing, stick with one terminal running one shell and using one private key, making connections in sequence.
