Will the Shaker also improve lung function similar to POWERbreathe?

The Shaker and POWERbreathe are two different types of devices that do different things. The Shaker is a mucus clearance device, rattling the mucus and catarrh and loosening it from your chest so that you can easily expel it.
POWERbreathe is a training device, training your breathing muscles to become stronger as you increase the resistance that your breathing against. So if you have a medical condition that results in excessive pulmonary secretions, the Shaker is the device you need. If however you have a breathing problem that means your respiratory system is weakened, or you simply want to strengthen it to improve your quality of life or sports performance, then POWERbreathe is the device for this.

What research has been produced to show the Shaker’s effectiveness in comparison to other devices (flutter)?

This is a useful link to a paper comparing the Shaker, Acapella and Flutter devices.

Here are other research papers that use and/or reference the Shaker. They are mostly from Brazil (and in Portuguese but the abstracts are in English) because this is where the product originates from: