Assume the character build Any Monk 5/Moon Druid 2, wildshaped into a Brown Bear:
On their turn, can they use their attack action twice to multiattack?
No. Extra Attack doesn't give you two Attack actions, it gives you two attacks per Attack action. A 5th-level monk still only has one Attack action.
That means that not only do you not get two Multiattacks, but if you take the Multiattack action you don't get a second attack from Extra Attack either, because only the Attack action triggers Extra Attack's benefits.
The Monster Manual (p. 9) says:
When a monster takes its action, it can choose from the options in the Actions section of its stat block or use one of the actions available to all creatures, such as the Dash or Hide action, as described in the Player's Handbook.
If Multiattack is one of those options; then the druid is choosing the Multiattack action, not the Attack action and does therefore not meet the prerequisites of any class that give Extra Attack.
If the Druid is currently a Brown Bear and wishes to attack on their turn they can choose the following actions: