Each beam is a separate attack in this case. So
Different from Magic Missile. Because each is a separate attack, it's almost like treating it as a separate instance of the spell. It's a separate damage roll to so you add your CHA mod to each instance (magic missile is one damage roll per target).
Yes. Again, because they are separate attacks, each beam you target at an opponent, that hits, triggers a new concentration check.
You're exactly right on the comparison, this is a multi-attack spell, rather than multiple damage units like Magic Missile.
Your Druid Gets Attacked, then the barbarian gets pushed
The trigger for your readied action was:
Cast Eldritch Blast (with Repelling Blast invocation) on the berserker in question if he attacks the druid.
It is stated in the Dungeon Master's Guide (p. 252) that:
If a reaction has no timing specified, or the timing is unclear, the reaction occurs after its trigger finishes, as in the Ready action.
So your readied Eldritch Blast will occur right after the berserker attacks the druid.
What if, rather, the trigger was the druid moving? What if the trigger was the berserker raising his weapon or rearing back in preparation for an attack?
The alternative triggers would not improve matters much. If you specified the trigger was "the berserker raises his weapon to attack", that might be tricky, because the trigger has to be a "percievable circumstance." In combat, everyone always looks like they're just about to attack each other: what if the enemy doesn't "raise" a weapon, but stabs with it? What if they raise a weapon in salute?
If the tirgger was the Druid's movement, that's no good either because the Barbarian will attack before the Druid moves away from his reach (as that's the timing specified in Opportunity Attacks), and you'd Eldritch Blast after the Druid finished moving away from his reach (since it's a Readied action, and happens after its trigger), so the berserker would still attack first.
A relevant question that has been asked before is can I specify a trigger to be "before... occurs"?
(Mostly, you can't).
Best Answer
The Ready action includes no restrictions on the number of attacks you can make. You might be thinking of the restriction in the Extra Attack ability:
So a character with Extra Attack can only make a single attack when they Ready the Attack action, but there is no general restriction on multiple attacks with the Ready Action.