In the absence of any context, I'd have to say your first assessment of the phrase is correct. To have something over someone else means you possess an advantage over them. In the context of a job interview, for example, that might mean you have a skill the other person does not.
To have something on somebody colloquially means you possess some information that could discredit them. This is very different. It is possible that the two phrases have become conflated.
One reason for possible conflation, as has been pointed out in comments, is that another unique specific phrase "to have a hold over someone" exists. This phrase means someone is controlling someone else. An example of this kind of control may be when someone uses damning information they "have on" the other person to bribe them, but there are certainly other ways this phrase is used, such as when a person has 'groomed' someone psychologically over time in order to exert control over them. It's worth noting that the phrase "to have a hold on" something also exists and similarly means you are in control, but not in a negative way - for example, you could have a hold on your finances meaning you are managing them well.
Another possible reason is that having something on someone could also be seen as having an advantage over them if used as a bribe, meaning that they also have something over them. In such a context, one is just a type of the other rather than something synonymous.
