Philosophers advocating a flat ontology are motivated by two different goals. First, they aim to accept as an object almost any candidate put forward. This results in a proliferation of many different kinds of objects, such as tables, chairs, atoms, strings, persons, cities, countries, representations, thoughts, feelings, qualities, events, facts, relations, and so on. Second, they desire to prevent any hierarchy between objects. No object should be more or less an object than any other object. In other words, there are no objects that are more fundamental or more privileged than others.
Both goals correspond to two different dimensions of any proposed flat ontology, namely its breath (scope) and its depth (hierarchy). The flattest ontology should have a maximal scope and zero depth. But then one given flat ontology can actually be more or less flat than another suggested one. Your flat ontology could be flatter than mine, just in case yours allows for more different types of objects or includes less hierarchical levels between objects.
Take quantificationalists such as Frege, Russell and Quine. According to them there are no things that do not exist. Saying this refers to the second dimension of flatness. For they do not want to introduce different levels of objects, that is to say, a hierarchy between objects that actually exist and objects that merely are but do not exist. However, with respect to the first dimension of flatness their ontology is in fact not flat at all. The reason for this is that they exclude many candidates for being an object. For example, they surely do not allow possibilia and impossibilia to be objects.
Contrary to quantificationalists, neo-Meinongians advocate a flat ontology in the first sense. For they want to count possibilia and even impossibilia as genuine objects, leading to a wide range of different kinds of objects. Yet, with respect to the second dimension they are in fact not advocating a flat ontology at all. The reason for this is that they accept a hierarchy between objects, namely between those that actually exist and those that just be but don't exist.
So neither quantificationalists such as Frege, Russell and Quine, nor neo-Meinongians are true advocates of a flat ontology. An example of a true flat ontologist would be Tristan Garcia. In the first part of his fascinating book Form and Object. A treatise on things he accepts almost* every candidate for being a thing, while at the same time preventing any hierarchy between things. Anything is a thing, no matter what. Everything is equally something. No more, no less.
(*) "The world is precisely that which is not some-thing. It hardly has any other determination" (T. Garcia, Form and Object, p. 86)