I'm a level 15 fighter and I have Extra Attack(2):

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

The number of attacks increases to three when you reach 11th level in this class and to four when you reach 20th level in this class.

I reincarnated as an Erinyes and on its features it has a Multiattack making three attacks.

Multiattack: The erinyes makes three attacks

So do I get 5 attacks now because the race grants me three attacks and my fighting style grants me two bonus attacks?

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.

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.

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 a different player-race. Monsters from the monster manual are generally 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 class are balanced against the other vanilla 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)