Le terme normal pour un cadeau est recevoir.

Avoir a de multiple sens:

• posséder : sens premier
• recevoir (un cadeau, une récompense, mais aussi des coups)
• obtenir (un diplôme, une récompense)
  …

Grosso modo, avoir permet de relier le sujet avec quasiment tout ce qui le concerne (situation, maladie, possessions, choses positives, choses négatives).

L’autre verbe générique est être qui lui se focalise sur l’état, la situation de la personne, ses caractéristiques.

Le passé composé j’ai eu, n’implique pas que cela est fini, seulement que cela est arrivé dans le passé (même très proche: il y a quelques minutes).

“Avoir” is linked to the concept of possession, be it of a material object or something abstract. For instance, you can say :

J’ai un ballon
J’ai 8 ans

because both a ball and age are considered a possession in French.

As for your question, you’re right, originally, j’ai eu meant “I had something but not anymore”.
But nowadays, when we use it without mentioning a specific date or period, it usually means something like ” J’ai reçu “/”I obtained/received/…”.
For instance,

J’ai eu une grosse opportunité.

usually means

I have a great opportunity.

Because nothing implies that I don’t have it anymore.
BUT,

J’ai eu une grosse opportunité à l’époque mais…

usually means

I have had a great opportunity at the time but…

At the time implies not anymore.

As you can see this is just for conveniance that we say it that way.
Unless you see a specific period of time, “j’ai eu” is to be understood “j’ai reçu/ j’ai eu depuis”.

To my ears using “J’ai eu x” here is like using “I got x” in English. In terms of meaning, it feels like it focuses more on the receiver, to the exclusion of the giver. In the context of children this makes sense because they can tend to be more “me” focussed. In the context of Christmas this even makes more sense because the “getting” can feel more magical and the “giver” is more removed and mysterious (Santa). So it feels more like “this is what happened to me on this magical day”.

Just my 2 cents.