There seems to be a confusion of terminology here. Fasting refers to the canonically required practice (for most adults who are not seniors) whereby food is limited by consuming one full meal and two small snacks; this is required only on Ash Wednesday and Good Friday. *Abstinence *refers to abstaining from meat on Fridays, something which in the U.S. is canonically required only during Lent, with the same abstinence or some other act of penance or devotion required in its place the rest of the year. Neither of these relates to the common lenten devotion of “giving up” something for Lent, which is customary, but not required.
As you may see, there are several variations of practice concerning this devotion, and none is the “right” or even necessarily the “traditional” practice. Some people give up the item in question for the duration of Lent, Sundays included, while others maintain that all Sundays being feasts, it is not appropriate to continue the penance on that day of the week. Some make exceptions to their discipline on two important days, St. Joseph on 19 March and the Annunciation on 25 March, both of which are solemnities, the highest category of feast on the kalendar. There is no universal practice in this regard, which might help explain why you are unfamiliar with it. Practices vary greatly by country, diocese, parish, household and individual.
I personally do not partake on Sundays, not because I do not recognize Sunday as the weekly feast of the Lord, but because I tried it once many years ago and found that the Lenten discipline was severely weakened, IMHO. It was not nearly the same sacrifice to give up something when I had weekly relief; I did not feel challenged. In other words, there was always a day, just a few days away, in which I could partake. OTOH, I do break the discipline on the two days above, because they are unrelated to Lent, and the church does observe them with music, Gloria at Mass, white vestments, etc… These are only two days, which sometimes get postponed past Easter, anyway, as happened last year. Other feel differently, but I found that the joy of partaking at Easter is sweeter when I have not partaken every Sunday.