Since I got no traction on this post, I ended up reaching out Atlassian support directly. They had the following response from the AI:
Currently, the Jira mobile app does not offer the same ability to call or email a user directly from within the app’s user directory, as was available in the Opsgenie mobile app. While many Opsgenie features—like alert management and on-call notifications—have been integrated into Jira Service Management and its mobile app, direct contact actions (call/email) from the user directory are not documented as available features in the Jira mobile app.
There is an ongoing feature request to address gaps in functionality between the Opsgenie and Jira mobile apps, such as the ability to create alerts directly from the Jira mobile app. While this specific feature request does not mention direct call/email actions, it highlights that Atlassian is aware of feature parity concerns. You can watch the ticket to get notified of any updates or developments.
Based on that, I requested that a ticket be generated to get this feature incorporated into the Jira mobile app. If interested, upvote and comment to show Atlassian this is a desired feature: https://jira.atlassian.com/browse/JRACLOUD-96684.