The Pawprint Shop is not a building: it is a giant Aniimo. The game says it appears in Astra Square on Saturdays. It has no icon on the in-game map; it stands right next to Horizon Square. The pin on the map is the wiki's own, placed where the game files put it. Walk up to it and pick "Trade" to spend Paw Coins.
Where it is
Astra
Markers currently shown on the map (45)
Landmarks - Horizon Square
Landmarks - North of Horizon Square
Landmarks - Northside
Landmarks - NPC (15)
Landmarks - Phoebe Street
Landmarks - Polaris Institute
Landmarks - Super Lucky Star - Voxel Coins can be used for a prize draw.
Landmarks - Tracking Point (3)
Landmarks - Transporter (2)
Shops - Aniimart - Transmog Shop
Shops - ChillBeanz - Coffee · Star-Crossed Tea Party
Shops - Coilstrike Club - Jaff
Shops - Eren - Star-Crossed Shop
Shops - FunTown - Astra FunTown
Shops - Ian - Mementos Owner
Shops - Jaff - Aniimo Training Item
Shops - Jiff - Catch Items
Shops - Lin Xia - Coffee · Star-Crossed Tea Party
Shops - Lumen - Aniimo Accessories
Shops - Mementos - Affinity Items
Shops - Pawprint Shop - Held Item · Training Materials