Learn Icelandic: Borðar einhver sömu sneiðina? - Is someone eating the same slice?

This is a special rule for the word sami (the same). It almost always uses its weak declension forms, and the noun following it must have the definite article attached. So in Icelandic, you are literally saying "same slice-the".

Why is the form sömu used here instead of sama?

sami must agree with the noun it describes in gender, number, and case. sneiðina (the slice) is a feminine noun and acts as the direct object being eaten, putting it in the accusative case. The weak, feminine, accusative, singular form of sami is sömu.

einhver means "someone", but why is it in the masculine form here?

When referring to an unknown person in general, Icelandic defaults to the masculine singular pronoun einhver. You would only use the feminine form if you already knew you were asking specifically about a woman or a girl.

Could I drop the definite article and just say sömu sneið?

Generally, no. When a noun directly follows a form of sami, that noun must be definite. Saying sömu sneið sounds unnatural to native speakers, though there are a few fixed phrases where the article is dropped.