In brief, the prompt is part of a new security feature Apple introduced with macOS 15 Sequoia.

For reference, an example of the prompt is included below:

In order to use your DisplayLink-based product, you must click the ‘Allow For One Month’ button within the prompt you received.  In one month’s time, you will be prompted to renew this authorization for an additional month. This renewal process will continue each month thereafter.

To expand further, when initially installed the DisplayLink Manager Application must be granted ‘Screen Recording’ permission within the macOS Security & Privacy settings in order to function.

*** An important note, the DisplayLink Manager Application does NOT in fact record or store any information. This permission must be granted in order for the DisplayLink Application to access the information it needs in order to generate the image shown on the DisplayLink driven displays. ***

With the release of macOS 15 Sequoia, Apple has instituted a new policy whereby this permission must initially be allowed for one month and then be renewed on a monthly basis. If the permission is not renewed, then the DisplayLink device will not function.