Bsrgvty

I'm playing a fighter with the Two-Weapon Fighting fighting style. Several times in the game tonight, I killed my target with my first attack, and so I wanted to use my bonus attack to go at one of the other monsters in the room. Once, I hadn't moved yet and so I used my movement to close with the second monster, and another time the second target was already in range without me needing to move.

Each time, however, our DM made me take a -3 to my attack roll on the second attack (he didn't even make me roll with disadvantage, just assigned an arbitrary numerical penalty) and said it was because I was switching targets.

I played a little of older versions of D&D growing up, but only really got into it a few years ago with 5e. Is this some old-school rule the DM is citing?

I've seen discussions online saying you can move between attacks, but haven't been able to find anything to suggest you take a penalty for switching to a new target.

This DM also isn't just capricious or vindictive – he always happily relents if we show him a rule in the PHB that goes against what he's saying, for example, and any homebrew/house rules get discussed beforehand. This is just the first time I've ever encountered this sort of penalty, and I've played with TWF with other DMs in the past...

There is no such penalty

The Two-Weapon Fighting section states (in its entirety):

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

Note that it never mentions any sort of penalty for switching targets and makes no requirement whatsoever that both of your attacks must target the same thing.

Additionally there is the question: "When attacking with Two-Weapon Fighting, can you break up your movement and attack a different target?" where it is explained that moving between your first attack and the bonus action attack is completely allowed by the rules.

I'm unsure if this was a point of confusion but know that any character can use Two-Weapon Fighting so long as they meet the prerequisites (wielding two light one-handed weapons). There is also a Fighting Style which the Fighter class gets called the Two-Weapon Fighting Fighting Style which states:

When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.

This allows you to gain a bonus to the bonus action attack by allowing you to add your modifier to it whereas characters without this Fighting Style only add their modifier if it's negative.

Note that as use @SeriousBri points out, there was a penalty in older editions to the second attack but there still wasn't one to choosing a different target for it.