For countries other than the United States, yes, SMS Sender IDs can be changed by country in Settings > Organization > Notifications > Sender Information (see screenshot to the right). You can add up to 20 custom Sender IDs per country.

NOTE: Custom Sender SMS IDs are only available for standard notifications without confirmation. Updating SMS IDs may impact Sender SMS ID settings for existing notification templates.

Custom Sender IDs are for display purposes only; Everbridge actually uses short and long codes to deliver SMS messages, a list of which can be found in knowledge article EBS: Everbridge Global SMS Codes.

Informing contacts of the short codes that deliver SMS notifications and encouraging them to add the short code to their device's contact list will increase response rates.