Take a spell like Stoneskin. One of my BBEGs (I'm a DM) cast Stoneskin on themselves to maximize their tanking capabilities. Our party wizard asked about Dispel Magic, and if it could be used to dispel the effect. In the end, it seemed RAW, so I let him. Though later, this seemed… weird. Stoneskin technically has 3 effects (resistance to 3 different damage types) and the target was the source of the concentration. Dispel Magic removes one magical effect, not one spell.
In the future, my personal ruling is you cannot end an effect if their concentration is on themselves. The wizard seemed surprised when I let him end it before, so I think he'll agree it seems reasonable. I'm comfortable with how I'll handle this in the future. I'm just curious if there are Sage Advices or Errata covering this that neither of us have seen.
Best Answer
Dispel Magic ends Stoneskin
Your original ruling was partially correct - here's how it should have gone down:
Dispel Magic states (emphasis mine):
The target creature currently has stoneskin active on it. This is a 4th level spell.
If dispel magic was cast at it's standard 3rd level, it would require a DC14 spellcasting ability check in order to end it per dispel magic:
If the caster of dispel magic knew that stoneskin was up and could cast it at 4th level, they could dispel it immediately.
The number of spell effects don't matter, it's the spell itself that is dispelled.
What about Concentration?
Concentration doesn't matter. What matters is the location of the spell effect and not who is concentrating on it.
What about multiple spells?
In the case of multiple ongoing spell effects, a single casting of dispel magic can remove all active spell effects on the target.