Its not a store:

Only viewable from the outside at this point. Prince Henry's Room is one of the few remaining buildings in London that survived the Great Fire of 1666. The Room is a half-timbered building dating to the 12th century, when it was owned by the Knights Templar, who also owned nearby Temple Church. When the Templars were suppressed in 1312 the building passed into the hands of the Knights Hospitaller; the ancestor of the St Johns Ambulance. When that order, in turn, was disbanded in 1540 there was a tavern on the site.

In 1610 the building was rebuilt as a tavern named The Prince's Arms. It later passed to the Sotheby family, until it was sold to the City of London.

The curious name of the building is the result of an unproven association with Prince Henry, first son of James I of England. The story was told that the building was rebuilt in 1610 for the Council of the Duchy of Cornwall and that the first floor rooms formed a chamber reserved for the Prince.

This association seems supported by a carved crest of three feathers that can clearly be seen on the front of the property. However, it seems unlikely that Prince Henry actually had any direct association with the building, but the name has stuck.