No. The Extra Attack class feature does NOT apply to Multiattack.
On your turn, you choose one action. Several options are defined in the combat chapter of the PHB, such as Attack, Dash, and Dodge. Additionally, some classes get special abilities that read, "As action, you may...".
Multiattack is NOT the Attack action. It's more like a special ability. During combat, a monster with a Multiattack listed in its stat block can choose Multiattack instead of taking the Attack action.
Look again at the Extra Attack ability:
Beginning at 5th level, you can attack twice, instead of once,
whenever you take the Attack action on your turn.
The text was deliberately written this way. The Extra Attack class ability does NOT apply to the Multiattack action. Jeremy Crawford (primary author of the PHB) agrees.
Monster stat blocks aren't written to be a playable race.
Even when a creature is a playable race (which the Erinyes isn't), the playable race version of a creature is different from that creature's Monster Manual stat block. This allows balance tweaks when a race might be too weak or powerful, but it also helps with the flow of the game.
As a player, you can spend lots of time customizing a character and calculating his or her stats. A GM running an encounter doesn't have the time or need to do all this customizing or calculating. The GM needs to be ready to let you fight a monster without having to do lots of prep work.
That's the point of a stat block. It's a typical specimen of the creature, not a customized character. With your figher Erinyes, not only are you customizing a character, you and your GM will have to customize the rules for what racial abilities this character gets.
Without bending the rules, it is actually possible to get both Multiattack and Extra Attack on one character-- and even then you CANNOT combine them. A character with 2 druid levels (Circle of the Moon) and 5 Fighter levels can turn into a Brown Bear with Wild Shape. Because Wild Shape allows the druid to retain its class features, the character may choose: EITHER do the Multiattack (bite/claw), OR do the Attack action with Extra Attack (and choose any combination: two bites, two claws, or one of each).