Translation:That girl there looks a lot like my sister.
I would not agree that to say "That girl there looks very like my sister." is entirely unnatural in English. I would normally tend to say, "That girl there looks very much like my sister." if I were to say this in conversation, but the first translation seems to me also quite plausible.
Yes, in most cases it sounds perfectly fine to use just "ligner" in place of the full expression, "ligner på". However, in this case it does not sound natural to me, and I suspect the reason is that there's an adverb between "ligner" and its object.
"å ligne" on its own actually means "to be similar", so then it's easier to understand the function of "på", as it gives direction, or a target, to the verb; points out the object:
"å ligne på noe"
"to be similar to something"
So you could say that "på" has the same function here as in cases where it would be translated as other English prepositions. It serves to give direction to, or a target for, an action:
"To think about X"
"To throw X at X"
"To write on X"
There is no 1:1 relation between the prepositions in English and Norwegian, and it's probably one of the last things a language learner is expected to master in either language, so your confusion is perfectly normal! :)