Learn German: Wischst du dem Hund den Dreck von den Füßen ab? - Are you wiping the dirt off the dog's feet?

This is the possessive dative (Dativus possessivus). Instead of using a possessive word like 'his,' German uses a dative noun or pronoun to show who is affected by an action on a body part or item of clothing. Here, dem Hund tells us whose feet are being cleaned.

Why is it den Füßen (the feet) instead of seinen Füßen (his feet)?

When you use the possessive dative like dem Hund, the body part only takes a definite article (den). Because the dative object already establishes who the body part belongs to, adding a possessive determiner like seinen would be redundant.

Could I use the genitive case here and say von den Füßen des Hundes?

Grammatically, yes, but it sounds very formal and unnatural for this context. For everyday actions involving body parts and clothing, German strongly prefers the possessive dative construction over the genitive.

What is the difference between wischen and abwischen in this context?

While wischen simply means 'to wipe,' the separable verb abwischen specifically means 'to wipe off' or 'to wipe away.' The prefix ab placed at the very end of the sentence emphasizes the removal of the dirt (den Dreck).