That blue box is a wired receiver unit for the wireless microphone (they currently use a Sennheiser handheld with SKP-100 wireless transmitter, but they did or still do use also products from other companies, inc. e.g. Shure that I know they flew during some STS missions to the station). The box also hosts its own microphone because the wireless microphone does need charging (EPO - Education & Public Outreach events usually last about half an hour or less, while they're in contact with CAPCOM throughout the working day on the station and the small battery pack it has wouldn't last that long without requiring a few battery swaps) and it might be awkward to use in some circumstances and not having to worry about a free-floating microphone might be easier too. For example, here's Karen Nyberg speaking into a microphone while conducting a session with the Advanced Colloids Experiment in the Destiny laboratory:
NASA astronaut Karen Nyberg speaks into a microphone while conducting a session with the Advanced Colloids Experiment
(ACE-1) sample preparation at the Light Microscopy Module (LMM) in the Fluids Integrated Rack/Fluids Combustion Facility, on
June 24, 2013. Image: NASA
The box has two large buttons that while I can't seem to find any supporting document to their purpose can really only be to toggle the wired and wireless microphones on and off and adjust their volume during communications with CAPCOM (Capsule Communicator):
CAPCOM communicates through the use of voice loops (collections of
audio communication from individuals which are tied together on an
intercom through a satellite link). CAPCOM also coordinates crew
communications with other spacecraft communicators supporting the
mission which may be in other locations - such as Huntsville, Alabama;
Munich, Germany; Tskuba, Japan or Moscow, Russia.
Here's a photo of Shannon Walker on the station showing the box with the two buttons and a microphone behind her a bit better:
NASA astronaut Shannon Walker, Expedition 24 flight engineer, is pictured near a robotic workstation in the Destiny laboratory of
the International Space Station. Credit: NASA, Source: Wikimedia Commons (click for higher resolution)
So in the video you link to, Samantha Cristoforetti would first have to reach for the receiver unit to enable both microphones to pass through the intercom voice loop anyway. But in majority of cases, having watched quite many public events from the station (available either live on NASA TV or recorded on NASA YouTube channel), station astronauts would mostly use the better quality wireless microphone.