My date of birth is the 28th of June 2008. I am Spanish (no origin)/ My parents are Spanish (wenn er die deutsche Staatsbürgerschaft hat). I like playing basketball and hearing music, but my favourite hobby is playing the guitar.

I'm in year 6 at the ... ('class' ist, zumindest in BE, der Klassenverband, nicht das Schuljahr)

In two months my class my class is/will be organising (ist ja fest geplant) a sports cup.

Ich finde auch 'My date of birth is' für ein Kind beim Vorstellen recht förmlich; ich würde einfach sagen 'I'm xx years old'. Aber vielleicht gibt es ja einen Grund für das genaue Datum

Du hast recht; es MUSS sogar 'listening to music' heißen, das habe ich übersehen, sorry. 'listen' heißt bewusst etwas hören bzw. zuhören; 'hear' heißt, dass ich halt etwas höre - das können auch die streitenden Nachbarn sein oder die Frösche im Teich.

At least in AE, one would rarely say "My date of birth is..." Instead, you'd usually hear "I was born on..." (You're likely to see "date of birth" on a form or somewhere where you need to enter the information, but it's overly wordy in typical writing or speech when a shorter form does the trick.)

(I know that goes against German's general dislike for starting a sentence with "ich"/"I," but it's not such a cardinal sin in English--you just want to vary your sentence structure so that you don't have a series of sentences all starting with "I.")

... is June 28th, 2008 ...

I'd say (AE) - for brevity's sake. :-)

Narrative form: I was born on ...

Or tabular form: Date of birth:

'Spanish origin' can mean all sorts of things. Be specific.