der Araber
Arab
A person who comes from an Arab country, is of Arab descent, or whose native language is Arabic.
Er ist Araber und lebt seit zehn Jahren in Deutschland.
He is Arab and has lived in Germany for ten years.

Arab / Arabian
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Arab
A person who comes from an Arab country, is of Arab descent, or whose native language is Arabic.
Er ist Araber und lebt seit zehn Jahren in Deutschland.
He is Arab and has lived in Germany for ten years.

Whether a person or a horse is meant can only be recognized from context. Words like 'reiten' (to ride), 'Stall' (stable), or 'Zucht' (breeding) point to the horse. Words like 'Herkunft' (origin), 'Sprache' (language), or 'Kultur' (culture) point to the person.
Mein Nachbar ist Araber und kommt aus Ägypten.
My neighbor is Arab and comes from Egypt.
The noun 'der Araber' (the Arab / Arabian horse) belongs to the word family around 'arabisch' (Arabic/Arab). This includes the adjective 'arabisch' (e.g., the Arabic language) and the geographical term 'Arabien' (Arabia).
Sie lernt an der Universität die arabische Schrift.
She is learning Arabic script at the university.
When 'Araber' (Arabian horse) refers to a horse, it is often combined with verbs from equestrian sports. Typical phrases are 'einen Araber reiten' (to ride an Arabian horse), 'Araber züchten' (to breed Arabian horses), or 'einen Araber besitzen' (to own an Arabian horse).
Sein größter Traum ist es, einmal einen echten Araber zu reiten.
His biggest dream is to ride a real Arabian horse someday.
Turk
'der Türke' and 'der Araber' describe people by origin, culture, or language, but they are sometimes incorrectly mixed up. A 'Türke' is a man from Turkey or with Turkish background. An 'Araber' is a person from an Arab country, of Arab descent, or with Arabic as a native language. Turkey is not an Arab country, and Turkish is not an Arabic language. Therefore, you should not use these words based on appearance or religion, but based on origin, language, or how someone identifies themself.

Der neue Kollege ist Araber aus Marokko und spricht Arabisch und Französisch.
The new colleague is an Arab from Morocco and speaks Arabic and French.
Mein Nachbar ist Türke und spricht mit seiner Familie oft Türkisch.
My neighbor is Turkish and often speaks Turkish with his family.