The Webex app client application is the intended replacement for Jabber with CUCM: it supports on-prem and cloud calling deployments. Webex is a brand name, not a single product at this point. The main reason Jabber still exists is to support air gapped environments, often with CMS. Note that the Webex app uses the newer cloud messaging architecture, if you enable it for a user, instead of IM&P XMPP.
The Webex app with UCM Calling reuses a lot of Jabber’s config: same device types in CUCM, same Service Discovery, same Mobile and Remote Access via Expressway. Unlike Jabber, it can also leverage newer features implemented in Webex Control Hub. The Cisco Call Microsoft Teams integration (config guide) is an example of this - and include server-side integrations such as the bidirectional presence sync @Roger Kallberg mentioned. The Cloud Connected UC architecture allows that integration between the Webex and Microsoft backends to include CUCM on the phone status in the presence sync. In fact, the exact thing you’re requesting shipped this month per the what’s new article:
Mute Cisco calls when you are in a Microsoft Teams meeting or call
You can now mute notifications for incoming Cisco calls when you’re in a Microsoft Teams meeting or call. You can also set up to decline the calls automatically.
Entitlement for the Webex app and Control Hub are included in on-prem Flex calling entitlement. If you don’t have access, ask your partner or Cisco account team for help.
