Is there a way to make phonebook avaiable for numbers in the same context?

I have several softphones in the same context. Is there a way for those phones have other phone numbers readily loaded in their phones without having to add numbers again every time they register a new device?

I don’t believe Asterisk has any native phone books, in which case you are either talking about something layered on top by a GUI, like FreePBX (https://community.freepbx.org/), or something in the phones themselves.

The only meaning that register has for Asterisk is the SIP REGISTER method. That associates a device with a pre-existing endpoint definition. There is no facility to trigger the provisioning of a phone when that happens, and in any case, it would depend on the phone model, and an unprovisioned phone wouldn’t know how to form the REGISTER request.