Two different Japanese condiments are called beni shoga. One is pink, flat, and sweet-and-sour beni shoga, which you see at sushi restaurants. Another one is red, julienned, and sour, but not sweet. Both of them are pickled ginger.
The pink sweet beni shoga is sometimes called gari at sushi restaurants, but English wiki calls it as sushi ginger as it is almost exclusively used for sushi. It is made of young ginger root harvested in summer.
In the picture left, lighter color root is young ginger root, which is harvested in summer. When the young ginger root is left in the ground until fall, it gradually loses moisture and becomes regular ginger (the slightly darker colored one in the picture) which is suitable to store for longer period of time.
The pink beni shoga is made of young ginger root, which is milder than regular ginger root. The root itself has natural pink tint, and when it is sliced and pickled with rice vinegar and sugar, it turns more pinkish.
The red beni shoga is made of regular ginger root. The root is pickled with red umezu, which is by-product of umeboshi production. After pickling, the root is julienned.
Most of the commercially available ginger products are now artificially colored.