What you propose cannot be done today using a single Republic number. To some extent, the ability is coming with a future iteration of Republic Anywhere: Texting App | Text from Computer | Republic Wireless. Today, Anywhere is limited to text messaging on one’s Republic phone and computers (Mac, PC or Linux). In the future support for Anywhere will be expanded to additional devices (iOS and Androids other than Republic phones). Plans to add calling are in the pipeline also. The one caveat is unless both phones are active, the second would be limited to WiFi. Any phone active on Republic requires its’ own plan. Two Republic phones cannot share a single plan.
The short answer is yes, however, more details regarding what you wish to accomplish might help regarding specific suggestions. For example, do you need both calling and text messaging? The classic mostly free alternative would be Google Voice. If something more robust with support is desired, you might look at Dialpad. Please know using your current Republic number would require transferring that number to the VoIP service selected, then reactivating one or both phones with new Republic numbers. Also know, Google Voice will not directly transfer a Republic number. There’s a convoluted workaround documented here:
Finally, I think you can probably avoid all this if the goal is simply to have a backup phone in case one goes south. An inactive Republic phone may be activated to replace a no longer working phone very quickly. If more detail is required on that it would help us better help you; if we knew the brands, models and (if applicable) generations of the phones in question.