(I cannot say if my answer will respond to your questions, which indeed, are a bit unclear).
If one browses through many-many economic papers, one will get the impression that "representative" just means identical. Indeed in large chunks of literature this is the case, for historical reasons.
The drive behind the adoption of the "representative consumer" modelling framework came from the Lucas critique on the previous-generation macro-models, and the requirement that macroeconomic models are "micro-founded". But true theoretical aggregation (with heterogeneity present) requires some skill and knowledge and the bulk of the discipline appeared to have quickly settled for the "representative means identical" framework.
The problem is that in such a case, you don't really have a macro-model, just a blown-up version of a micro-model (note: here the words micro/macro are not to be mapped to the partial/general equilibrium concepts). There is nothing to "aggregate" here: the whole point of aggregation is to see whether the behavior of the collective differs from the behavior of the individual. And in the "representative means identical" approach, no such thing can happen, by construction: Instead of micro-founded macro-models, we ended up having blown-up micro-models posing as macro-ones (this is not opinion, I am just describing).
There are some models where the term "representative" obtains some intuition -especially in models with more than one classes of agents (say, labor owners and capital owners). Here we model two agents, and each is "representative" of its class. In-class are all identical, but here it sounds more appropriate to call the two individuals "representative".
The funny thing is, the concept "representative consumer" (RC) does have a special meaning: the representative consumer represents all consumers as regards basic structure, not measurement or quantities. E.g. "all individual maximize utility from consumption ("same structure"), but their utility parameters may differ("different measure"). All consumers have wealth, but the level of wealth may differ. Etc. RC is still a modelling abstraction, but it does leave room for heterogeneity.
A good source on the matter is
Caselli, F., & Ventura, J. (2000). A representative consumer theory of distribution. American Economic Review, 909-926.
Apart from their focus on developing a theory of distribution in the context of an RC model, they make a good job in presenting what can be done in an RC framework and what not. An excerpt:
The RC is a fictional consumer whose utility maximization problem when
facing aggregate resource constraints generates the economy's
aggregate demand functions. The RC assumption does not rule out
consumer heterogeneity, but only requires that potential sources of
consumer heterogeneity have sufficient structure to ensure that the
sum of all consumers behaves as if it were a single consumer.