No.

You quoted the relevant text. Extra attack specifically only triggers when the Attack action is taken, not on any attack. By contrast, the Multiattack racial feature is an action that allows you to take three attacks.

Thus, because the Multiattack action is not equivalent to the Attack action, Extra Attack does not trigger. Additionally, you do not get two Multiattacks, as is described here.

An Erinyes is a monster, not a race

This means two things: one is that you're not actually supposed to be playing as one, and the other is that if you REALLY want to be playing as one, you shouldn't use its stats as-is on top of your existing class features.

The distinction between monsters and PC races is not arbitrary. Mechanically, monsters are designed for a very different role than PCs. They are generally intended to have lots of power for a single encounter and not that many long-lasting buffs, while PCs are geared towards longer-term stamina - taking out several encounters a day, not just one. Monsters are not usually expected to survive the combat encounter, so their focus on single-encounter scale is understandable.

To better understand the issue, you can look up races that exist both as PC races and monsters. There's a big difference in how the two "interpretations" of the same race work.

Even if you choose to use the Erinyes stat block for your character (your GM can - and, if you ask me, should - disagree) you cannot stack your normal extra attacks with the Erinyes' multiattack. Extra attack only kicks in when you use the Attack action. Multiattack is a separate action of its own that happens to work similarly.

You have to ask your DM

We can't help you with this because Reincarnate lists all of the available race options and not only is the Erinyes is not one of them, but none of them have multiattack.

Because we do not know the details of the houserule your DM has used allowing it, we can not answer how those will work. There are general rules that cover multiattack and extra attack, but whether or not those apply to this particular houserule is entirely up to your DM and can only be answered by them.

You are very deep in homebrew territory here. While there are rules for reincarnation, those assume that the character reincarnates as one of the standard player races. Monsters from the monster manual are not designed to be played by players. They can often do things a PC could not do without a feat, magic item, spell or class level. Some monsters can do things which would be completely off-balance if used by a player.

Monster manual entries also throw a lot of things together which would usually be class features, perks from magic items, spell effects or feats if a player-character would have them (multiple attacks usually come from the class). It's hard to tell what's actually a biological feature of the monster's species, what's acquired through their (imaginary) character level and what comes from the equipment they usually carry around. Most of the time it doesn't matter anyway. But it also means that you can't just take a monster from the monster manual and convert it to a player-character.

So if you want to permanently play an Erinyes for flavor purposes, then I would recommend to forget about the monster sheet and instead home-brew Erinyes as a new player-race. Make sure that any special rules for this race are balanced against the other standard player-races from the players handbook. You might notice that none of them grants you anything which is nearly as powerful as raising your regular damage output by 150%.

But if you are going for a "rules as written" interpretation, the rules say "The erinyes makes three attacks". It does not say "two additional attacks in addition to its normal attack" or even "three additional attack on top of any attacks it gains from some other source", it says specifically "three attacks". So if you ended up temporarily playing an Erinyes due to some magic shenanigans you would always attack three times.

(One such method, True Polymorph, explicitly states that you do not retain any mechanical features from your previous form except your alignment and personality. So your two additional attacks from Lv.11 fighter wouldn't apply)

If you retain your Fighter class features: Yes, but not with the Extra Attacks Fighter class feature. Instead, you could gain extra attacks by using the Action Surge Fighter class feature.

When you use the Action Surge Fighter class feature, you can take an additional action of your choice on your turn, on top of your normal actions. Since the Erinyes's Multiattack is an action, this means that you can use the Erinyes's Multiattack action, then Action Surge and use it again, giving you six attacks.

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.

Fun Fact

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).