Yes, you can use Osmo with multiple devices.
If you own a Starter Kit, you can also purchase one or more extra Osmo Bases so multiple players can use Osmo games at the same time.
All paid (digital-only) apps can be accessed on multiple devices as long as you are signed in to the same Apple or Amazon ID and myOsmo account as those you used when purchasing the apps.
You can set up Osmo on a new device following these setup instructions.
You can download any Osmo app using the steps described in this article.
To connect a new device to your myOsmo account, please open any Osmo app and follow the instructions to log into your account (and also create an account if you don't already have one).
For more information, please see our PDF guide to myOsmo.