A positive word for someone who is viewed like a sister, but isn’t one

You could try surrogate sister. Surrogate, in this case, would fall under this meaning:

surrogate
noun

1- A substitute, especially a person deputizing for another in a specific role or office.
she served as a surrogate for the President on a trip to South America

It then has the connotation as someone acting as a sister but not necessarily being one.

You could use virtual sister, The Cambridge Dictionary defines virtual as follows:

almost, but not exactly or in every way:

She was a virtual unknown before this movie.

Snow brought Minneapolis to a virtual standstill yesterday.

Using the adverb might sound better, e.g. "she was virtually his sister".

perhaps you could use "soul-sister" - according to the very trustworthy Urban Dictionary a "soul-sister" is :

"Someone who fully understands you. Like a soulmate, but not someone you want to marry and make babies with. The sister of your soul."

Another option is "sisters by heart". It implies that there is no blood relation between you two, but you're close, like sisters.

I’d go with honorary sister. I have a friend who is like a sister and I call her big sister, or MHBS - My Honorary Big Sister. She calls me little sis.