You can create a client account usin the same freelancer Upwork account. You can't have more that one account for the same person.
Your partner can indeed also create just a client account and invite you, or you can activate your Client account in your freelancer accont and then invite him. Both will work.
If you're a freelancer or agency manager now, but would like to hire others as a client
- Go to Settings › My Teams and choose the Create New Account button at the bottom.
- Choose New Client Account
- Once set up, use the account menu at the top-right of your screen to switch between using Upwork as a client, freelancer, or agency
Any questions, let me know!