In Savage Worlds a Double Barrelled Shotgun has a Rate of Fire of 1 or 2. Under the descriptive text is says that you can fire both Barrels together and roll to hit twice. One of my player wanted to know if he can fire at two different targets instead of both at the same targets. Weapons that have Auto Fire can at increasing penalty up to there RoF. But a Double Barrelled Shotgun does not have the Auto Fire feature. I have no experience of a real Double Barrelled Shotgun so I have no idea if it should be able too. My players think so. I think the rules as written say you can't but I am not sure that this is what they intended or it is correct IRL.
Yes, you can fire at two different targets with each barrel.
Any ranged weapon with a RoF of greater than 1 can be fired at two separate targets. ("Ranged Attacks, Rate of Fire", p. 67, Savage Worlds Deluxe edition)
The double-barrel shotgun is special in that it has this additional rule (page 50):
This means that unlike other RoF 2 weapons, it can use the Automatic Fire rules (p. 70), which are normally only available (and are mandatory for) weapons of RoF 3 or more. The "Double Barrel" rule doesn't restrict a double-barreled shotgun to only using the Automatic Fire rules, it just gives you an extra option to access the AF rules by "fir[ing] both barrels at once".
So there you go: the rules easily allow for using the barrels separately, one at a time, on two targets. Like many Savage Worlds rules, this follows from default rules, but it's not explicitly said about the shotgun. It's easy to read the special notes under Shotgun on page 54 and assume that this overrides the normal multi-attack rules, but it's in addition, not in replacement of, those rules. Go forth and double-barrel those multiple targets!
A tangential note
A shotgun-user shouldn't overlook the Innocent Bystanders rule (p. 73). This is really very tangential to your question, but it relates to how we intuitively expect shotguns to work with multiple targets and is easily overlooked.
This rule means that when you miss with a ranged or thrown weapon (and it's dramatically-appropriate in the GM's judgement), you automatically hit a target adjacent to the original. With a shotgun, this happens instead on a 1 or 2, because they're just so wide-effect. It means firing a shotgun into a cluster (of, say, zombies) makes it very likely you'll hit something – just make sure that other something isn't a friendly who's engaged in melee with your original target! This is especially effective when using the Double Barrel rule, since you might miss with one and hit with the other, allowing two targets to be "hit" at once.
As a side note to this side note, by the rules any shotgun, even one firing non-spreading slugs, will hit a bystander on a 1 or 2 when using this rule. I would personally rule as the GM that the extra chance on the 2 only applies to regular shot, because I think it's an oversight in the rules and not intentional to have it apply to slugs. As the GM it's up to you in your own game, but I wanted to highlight that as something to consider.
I don't have access to the rules, but in general, especially with respect to the real life application, a double barrelled shotgun can only be fired at to specific targets simultaneously, if they are really close together. Even then, I would rule it more as a colateral damage and not as a result of specific targeting.