Yes, Escaping From Entangle is an action.
From the Savage Worlds Deluxe Explorer's Edition source book's description of Entangle on pg 130
Each following action, an entangled target may make a Strength or Agility roll to break free.
And actions are specifically defined on ibid pg 74
Characters perform “actions” when their card comes up each round. A character can move (see Movement) and perform one regular action—attacking, running, casting a spell, and so on—without penalty.
I can see no support for the Shaken interpretation of the rules. If they wanted to add Entagle / Grapple / Ensnare to the Shaken update, they most likely would have. The Shaken update's scope is very narrow and well defined. It only applies to shaken rolls, and nothing else. From the Savage Worlds FAQ and Update May 2015
On their action, Shaken characters must attempt to recover from being Shaken by making a Spirit roll:
• Failure: The character remains Shaken. She can only perform free actions.
• Success: The character is no longer Shaken and may act normally.
There is however, some support for the grapple interpretation of "a raise allows you to act on the round you escape". This is how grapple works (as you noted) on ibid pg 83,
Once entangled, the defender may attempt to break free on his next action. Both the defender and attacker pick either their Strength or Agility and then an opposed roll is made. If successful, the defender is free but the attempt consumes his action. If he does so with a raise, he’s free and may act normally.
Also there is this rule from the Savage Worlds Super Powers Companion's description of the Ensnare power on pg 27 - which works almost exactly like Entangle.
[T]he target may make a Strength or Agility roll at –2 to escape. A raise allows the victim to break free and act in the same round.
Of course the other option is to just declare two actions before trying to escape the Entangle. But, if you allow raises to "[allow] the victim to break free and act in the same round" that opens another can of worms as the game is not clear on how it handles a conditional action economy.
What you might do is have them declare two actions, and if they escape with a raise, make them immediately declare an action to occur after their currently declared actions. This action will not cause any additional multi-action penalties.
There are of course many ways to handle this. But no matter what, it is likely best to decide now how your group wants to handle it so you will all have the same expectations going forward.