Learn Polish: Lubię to, co daje mi anonimowość w internecie. - I like what gives me anonymity on the internet.

In Polish, when 'what' means 'that which' or 'the thing that', you usually need the demonstrative pronoun to paired with the relative pronoun co. You literally say 'I like that, which gives me...' (Lubię to, co daje mi...). Just saying Lubię co... sounds grammatically incomplete.

Why does anonimowość look like it's in the nominative case when it is the direct object?

It is actually in the accusative case here, as the direct object of daje (gives). However, feminine nouns ending in a consonant, such as those ending in -ść, have the exact same form in the nominative and accusative cases.

Could I put mi at the end of the sentence, like daje anonimowość mi?

No, that would sound very unnatural. Mi is a short, unstressed pronoun. These typically tuck in right after the verb or near the beginning of the clause, but avoid the very end of a sentence. If you wanted to strongly emphasize that it gives anonymity to me (and not someone else), you would use the long, stressed form mnie.

What is the rule for the spelling change in w internecie?