In this phrase, 'that' is being used as a pronoun, standing in for the noun 'the vote'. So perhaps it could be read as:
Indeed, in the first round he got a vote 80% of the vote of Emmanuel Macron's.
In less terse, more readable wording, one might say:
Indeed, in the first round, the votes he received totalled 80% of the votes received by Emmanuel Macron.
Edit: to answer your question in comments about whether the original sentence is proper English, my opinion is: just barely.
It reflects a style choice by The Economist to use as few words as possible, and makes the sentence difficult and awkward to read, even to a native speaker of English. It's not incorrect, but it's really not great. An editor from another magazine (or an editor paying more attention) would probably have rewritten it to be more like my last example above. Even a simple change to:
Indeed, in the first round he got a vote which was 80% that of Emmanuel Macron's.
would have made it much easier to read, and more correct.