The dessert called sundae in US is mostly known as parfait(however pronounces as PA FE). It is very common western style dessert in Japan where you can find everywhere.
Standard size parfait comes with tall glass full of ice cream, whipped cream, fruits and other toppings, and the price can be similar or even higher than the normal main dish such as ramen noodle or curry with rice. Due to Japanese do not have custom to eat dessert after their main dishes, people usually order parfait in the afternoon as a snack.
Also it is very hard for Japanese men to order the parfait due to Japanese consider as men drink alcohol but do not eat sweets. Therefore older generations are reluctant to order parfait at cafe when he is alone. However, the life style is changing and now more men eat sweets than before.
Japanese style parfait is definitely with matcha ice cream. Green tea flavor is not an authentic Japanese ice cream in Japan but it is funny that major ice cream manufacturers do not sell match flavor ice cream a lot. There is no matcha ice cream bar or cone you can find in convenience stores. On the other hand, azuki red bean ice cream bar is a very typical one you can find anywhere in stores.