Very unlikely to have battery interoperability across brands. This is intentional and by design, as the manufacturers don’t want to make it easy to escape their ecosystem.

You’re only into the CyberPower for $40, chalk it up as a learning experience, and buy one of the many used APC units with wider availability of replacement batteries.

If you are US-ian, this is my favourite replacement battery vendor:
https://excessups.com

Depends if it uses weird connectors. Some UPS use normal F terminal connectors and some UPS battery packs like APC occasionally use plastic slide in trays that hold batteries and have goofy proprietary connectors. The batteries inside the sled can be replaced with the normal SLA batteries that are commonly used in UPS units though, just reuse the sled.

Yes. The Cyberpower's tend to use a 9AH battery in the same form factor that's normally a 7AH, but it's just a higher capacity. Check what physical size the UPS uses and go from there. Keep in mind you can have different battery sizes for the same capacity UPS. APC used to make a 280, 420 and 650 that all used the same batteries, and a 650 that used the next size up. All appeared identical except for the cutout in the bottom where the battery was loaded in.

Did you give the CyberPower a good, long time to charge the new battery? UPS chargers tend to be small.

Regardless: there's a decent chance that the new battery you purchased will fit an APC UPS. In this capacity range, a lot of UPS use a 12 Volt battery, of about 7 - 7.5 AH capacity, with physical dimensions of 6 inch x 3 11/16 inch x 2 9/16 inch, with push-on quick disconnect tabs. If that's what you've got, look for a used UPS that takes that size replacement.

(That's what I put into my CyberPower 900VA UPS - but it would have fit an old APC 600VA unit I have, just as well.)

(For that matter: an automobile jump starter I just repaired used a battery of the same size - but with different style terminals.)