After I have stayed there for two weeks, now I find myself pretty disappointed.
The acceptance of a credit card is incredibly low in Taiwan.
Restaurants and cafes: arguably, 95% of them don't accept the card. Most of those that accept it are ones that charge you with a service fee (usually 10%). The price per meal starts with about TWD 200.
One day I tried 15 restaurants and cafes located in Ximending, but found that no one accepted a card at all. I finally found 16th restaurant within a mall, but they took 10% service charge.
The other day I walked around Xinyi, but all of about 6 to 9 restaurants and bars didn't accept it. Even a bar that costs TWD 400 in one meal and drink didn't accept my card...
Starbucks accept all of your cards just like it did in other countries (including China). Dante Coffee accept it, but not Amex.
Souvenir shops on the street: Some of them accept the card in Ximending. I'm not sure about its probability, but it seems that at least 30% and up to 95% accept it. For example if you want to buy a Pikachu doll or pineapple cakes in sourvenior shops in Ximen street, you can use your card.
Convenience Stores: They don't accept any cards. However, you can use a transportation card (悠游卡), which I used in 7-11.
Night Market: Almost all of them don't accept it. Even those restaurants that have a store, insted of a stand booth, won't accept it.
Public Transport: You can't use a card to purchase a transportation card (悠游卡) nor top it up. I don't use a taxi, but at least you can use your card at Uber.
As to the brand, either Visa or MasterCard has likely the highest acceptance, followed by JCB. Amex is quite limited, more limited than in other countries, which tend to accept Amex more than JCB, I believe.
UnionPay is sometimes accepted especially in places many tourists visit, as well as AliPay, but the acceptance is more limited than JCB, at least guessed from a sticker on the door.
The most shocking experience is that I couldn't use my card at the Songshan airport, to purchase a SIM card. I didn't have cash at that time, so I just left the airport without mobile connection. However, one branch in the city accepted my card (Amex), but anyway it made me pretty disappointed to have no mobile connection from the airport (unlike mainland China, a metro in Taiwan doesn't have Wi-Fi available).
There seems also a mobile payment service called Gomaji Pay, but I didn't find anyone use it. So many people here just pay everything by cash when eating something, even in a restaurant.