So, I'll lay out some facts before writing down my conclusion:

I noticed that Emilia is NOT just a half-elf with silver hair, she is exactly identical to Satella, according to Subaru; Meaning, that her relationship with Satella is such that she is bound to be identical to her.

Echidna hates Emilia. Echidna also hates Satella. Echidna is emotionless, therefore hating someone must be very special and unique, otherwise her being emotionless would make no sense.

When one absorbs a Witch factor, their thoughts and goals don't change - they just become crazily twisted; This means that Betelgeuse ALWAYS "loved" Satella. That means that Betelgeuse ALWAYS wanted to bring her back.

As a child, Emilia had BETELGEUSE and Fortuna around.

Cloning is a thing, as shown with Ryuzu.

Let's now get to the conclusion:

Being perfectly identical to Satella, Emilia can only be Satella's clone, rationally speaking.

Having been raised by people who wanted to bring Satella back, we can assume that Emilia is an existence made for bringing Satella back.

Emilia's mother, which Echidna knows, is in fact Satella - but not because she got pregnant and gave birth to a child, just because they are clones.

Emilia is Satella's clone, confirmed.