Learn Russian: Эта упаковка такая же, как оригинал. - This packaging is exactly the same as the original.

We use такая же because it acts like an adjective describing the noun упаковка (packaging). Since упаковка is a feminine noun, we use the feminine form такая. You would use так же (which is an adverb) if you were comparing how an action is done, rather than comparing two objects.

Since оригинал is masculine, shouldn't we use the masculine такой же?

No, because такая же agrees with the subject of the sentence, which is эта упаковка (a feminine noun). You always make такой же agree with the first item in your comparison, not the second one.

What does the word же actually do in this sentence?

The particle же adds the meaning of 'exactly' or 'the very same.' If you just said такая, как оригинал, it would mean 'such as the original' or 'like the original.' Adding же emphasizes that it is 'exactly the same as' the original.

Is the comma before как strictly necessary?

Yes! In Russian, you must always put a comma before как when it is used to make a comparison like this. This is a strict punctuation rule that differs from English.