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No. PlayStation 2 does not have PlayStation Store.
In order to connect to the Playstation store you must sign in to the Playstation Network and select the icon from the Xmb for the Playstation Store.
It is not available on the PlayStation Store.
PlayStation Store was created in 2006-11.
There are many digital products available for purchase from the PlayStation Store. The PlayStation Store has games and movies, hundreds to chose from.