Similar to another comment, I think it's because "looks like" is too confined to visible characteristics, whereas "similar" could refer to a wide range of characteristics, such as personality, hobbies, etc. In addition, "looks like" is translated to the separable verb "aussehen." :)
Some verbs have two objects, a direct object in the accusative case and an indirect object in the dative case. The order of these two objects in a sentence follows these rules:
1) If both objects are nouns, the dative object will be placed first, the accusative object second: Ich schicke meiner Mutter eine E-Mail.
2) If one of the objects is a pronoun and the other is a noun, the pronoun will always be first. Ich schicke ihr eine E-Mail. Ich schicke sie meiner Mutter.
3) If both objects are pronouns, the accusative object will be placed first, and the dative object is placed second. Ich schicke sie ihr.
The above is sourced from the Menschen A2.2 Deutsch als Fremdsprache Glossary XXL book. This can be seen in the free sample pages from the book that Hueber, the publisher, provides at: https://shop.hueber.de/en/menschen-a2-2-gloss-xxl-dt-engl.html?___from_store=de#musterseiten
Click on "Leseprobe" in the sample pages section, and scroll down the PDF to page 17.
Here's some additional info:
The verb is just ähnlich, not "ähnlich sehen". There is no mention or comparison to "looks" in this lesson, just "similarities": ie, personality, mannerisms, temperament, attitude, interests, dress, as well as appearance. But, it isn't specifically noting that they "look alike", just that they are "similar" to one another.
ähnlich can be an Adverb, Adjective, or a Preposition. In this instance, it is used as a preposition, but in order for ähnlich to be a preposition, it must include a Dative case ("dir"). This makes "ähnlich dir" = "similar to you".
But, in regards to using "zu" in this sentence, I don't believe it has a place because it's redundant:
ähnlich (+dative) = "similar to"
dir = "to you"
"ähnlich ZU dir" would be something like, "similar TO to you", thus making "zu" useless.
That sentence is fine, but has a slightly different meaning - "the same" is much more like "=" than "similar" is.
Two bouquets of flowers, each with 12 yellow roses, have the same number of flowers and the same color and type of flowers. If someone took a flower from one of them, then the two bouquets don't have the same number of flowers, but they are still similar: they still have the same type and color of flowers.
Good question! I initially thought it would be "wie du" because the person being spoken to isn't the object, in that case they wouldn't need accusative. I guess if the sentence was slightly different, eg. "Deine Schwester sieht dich" then it would be appropriate to use accusative. Again for me it's just what I think but not to sure. If a native speaker could clarify this, it would be great :)
"Alike" and "similar" have different meanings.
You write "ähnlich" with a silent 'h' and "similar" with 'i' and not 'u'.
And yes, it follows the same pattern:
"Deine Schwester ist mir ähnlich." = "Your sister is similar to me."
"Deine Schwester ist dir ähnlich." = "Your sister is similar to you."
"Deine Schwester ist ihm ähnlich." = "Your sister is similar to him."
"Deine Schwester ist ihr ähnlich." = "Your sister is similar to her."
"Deine Schwester ist uns ähnlich." = "Your sister is similar to us."
"Deine Schwester ist euch ähnlich." = "Your sister is similar to you (plural)."
"Deine Schwester ist ihnen ähnlich." = "Your sister is similar to them."