The girl said, "Oh I thought Danny was giving me his scarf because it was so special, but he gave me it because he doesn't support United any more!"
Why past continuous here? Is it because it is a regret? It is not a continuous action, it is completed because the girl has the scarf.
We use the past continuous to describe a scene in the past. — It was raining. The wind was blowing. —
We use the past continuous to describe an event that was in progress at a specific moment in the past. — At 7:30 this morning, I was waiting for the bus. —
We use the past continuous and the past simple together to describe a sudden event that interrupted a longer one. — While I was having lunch, my cousin phoned.
This is easier to understand if you think of how this would be expressed in the present tense.
In Present-day English the present continuous is often preferred to the simple present in speaking of unique, individual actions because the simple construction is felt to imply repeated, habitual practise. Thus, if the girl were speaking of the gift as it was occurring, she would not say:
Danny gives me this because it is so special.
although that is entirely acceptable, and three or four generations ago might even have been the most natural way of saying it. Instead, she would say
Danny is giving me this because it is so special.
In your example, she uses the continuous form because she is recalling what she thought at the time; in effect she is 'quoting' herself. The past continuous is a 'backshifted' present continuous. It is only when she realizes her mistake after the fact that she shifts to the simple past, which as you say expresses a completed action.
This sentence is a little weak on grammar, though it does still sound like spoken English, where grammar can be estimated and not seem strange unless it is written down.
If I had written this, it would have been: The girl said "Oh! I thought Danny had given me his scarf because it was so special, but he gave it to me because he doesn't support United anymore!"
"Was giving": in the past, had been in the process of giving. Brings to mind what he was thinking while he did the action, and that it was the action in process that is important.
"Had given": in the past, has already been done. Brings to mind what she thinks just after she was given the scarf.
