It would be great if the ArcGIS server has the capability of publishing multiple mxds that stored in a folder by right click then publish all.
We have some recent workflows available which may be helpful options for you - They don't exactly address the request being asked (being able to batch-publish .mxds from a folder in ArcCatalog), but represent more modern approaches to this.
First up is bulk publishing with ArcGIS Enterprise. This workflow allows you to add a data store item to your Enterprise portal, and easily publish the content from your data store in bulk. Some more information about this can also be seen here: What’s new in ArcGIS Enterprise 10.7.1
Next, I encourage you to explore options when publishing to your server/Enterprise through ArcGIS Pro. When using ArcGIS Pro, it is possible to share a web map which automatically publishes the underlying layers as services - This is a more robust workflow than what was previously achievable through ArcMap, and you may want to explore this further.
I hope this helps!
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