There is no OOTB option to move a web part from one page to another page in the SharePoint online. You only could move and duplicate a web part in the same page.
Here are two workarounds:
1.On the top bar click the plus + next to "New" then select "copy of this page" allows you to make a complete copy.
2.You could develop a custom SPFx quick links web part with packaged links.
https://github.com/clarktozer/spfx-quicklinks
Meanwhile, to get this option in the SharePoint online, I recommend you give suggestion in the feedback portal and help to make improvements. Many features of current products are designed and upgraded based on customers’ feedback. With requirements like this increase, the problem may well be released in the future. Thanks for your understanding and support.
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.