Does the MiniBridge option in Photoshop needs internet connection to work?

No. it's just an integrated version of the Bridge into the Photoshop.

No need for internet connection, it's an icon link to open the bridge.

No. its  Bridge into the Photoshop

Mini Bridge is an extension that lets you work with assets in a panel in Adobe® Photoshop®, Adobe InDesign®, and Adobe InCopy®. It is a useful way to access many Adobe Bridge features when you work those applications. Mini Bridge communicates with Adobe Bridge to create thumbnails, keep files synchronized, and perform other tasks.

Open the Mini Bridge panel by doing any of the following:

• (Photoshop) Choose File > Browse In Mini Bridge.
• (Photoshop) Choose Window > Extensions > Mini Bridge.
• (InDesign, InCopy) Choose Window > Mini Bridge.