The snazzy new Google Hangouts Chat seems to support formatting links with alternate text like this one using this syntax
Source: https://developers.google.com/hangouts/chat/reference/message-formats/basic
However, their web UI does not give the <URL|text>
syntax any special treatment:
Is the syntax to format links in the UI is different than the API, and if so, what is it?
The documentation you linked is only relevant to calls made via API or webhooks. Only limited message formatting is offered via the Web UI and it does not appear that URL formatting is included.