Learn Spanish (Latin America): Trabaja igual que su padre.

While como translates to like (Trabaja como su padre), using igual que adds emphasis. It specifically means exactly like or the same as. It highlights that their way of working is identical, not just similar.

This lesson is about clausal comparatives using de lo que. Why does this sentence just use que?

The de lo que structure you are learning now is for comparatives of inequality (more than / less than) when a conjugated verb follows. This sentence is a comparative of equality (the same as), and it is followed directly by a noun (su padre) rather than a clause with a new verb. Therefore, a simple que is all you need.

Does igual need to agree in gender and number with padre?

No. In this sentence, igual is functioning as an adverb describing the verb trabaja (how he works). Adverbs are invariable, meaning they never change their endings to match the nouns around them.

Can I use lo mismo que instead of igual que here?

Absolutely. Trabaja lo mismo que su padre is another very common and natural way to express this exact same idea in Spanish.