Speaking as a legal professional licensed to advise in your jurisdiction and as a representative of your employer, despite being equally unqualified for both, the answers to your questions are
- No*
- Maybe
Copyright law varies across the many legal systems of the world and is complex and nuanced, however it concerns ownership of created works. Installing and using software to store and access passwords seems unlikely to create any works, but even if it did, then the ownership of the work would almost certainly not depend on the software or the computer, but the contract with your employer - particularly if you agreed that any work created is considered work for hire.
See e.g. Chapter 11: Subject Matter and Scope of Copyright of Title 17 of the United States Code.
There may be an issue with you installing software on your employer's computer. It's their property and they chose how it should be used, the details of which they should clearly communicate to you in the form a policy such as terms of use.
You appear to focus on the license which exists as a result of the contract between you and 1Password. Licensing of software on company computers is almost always a matter of concern to the employer, though the concern is usually related to liability arising from the use of unlicensed software.
Given my initial disclaimer and the lack of any relevant detail in your post, this reply is provided as-is with no warranty of fitness and a 110% money back guarantee.
*[As in maybe.]