Safety Considerations for Wired Bathroom Cleaning Electric Brush
For safety, the important information is on the back of the product and impossible tor read. I can see the box in a box symbol the means that it is double insulated. There should also be something to indicate it conforms to applicable safety standards. As mentioned in another answer, Residual Current Disconnect AKA Ground Fault Interrupt protection is important. You should not need to add overcurrent or overvoltage protection
You have not asked about compatibility with the motor. That is likely to be a problem. A motor requiring 4-5 amps to run requires much more than that to start, perhaps as much as 50 amps. The initial current will likely last only a fraction of a second, but that will likely be enough to make the power supply shut off for overcurrent protection. You would be better off with a simple unregulated 5-volt power supply that has a high tolerance for the very brief but high pulse of current that the motor is likely to draw when it starts.