Just to add to what /u/itsallfolklore said about stories in general, one of the main reasons to assume that Greek belief in their gods was genuine is the fact that the gods don't just appear within a discreet set of texts we can label "myths". They were everywhere.

When we talk about Marvel characters, we are referring to the specific comics and movies in which they play a part. We understand their role purely in terms of what their authors make them say and do in their story lines. They don't step outside these stories and become powers that we believe exist in the real world and affect our daily lives. They remain fictional and live within their own universe.

This is not true for the Greek gods. While the Greeks told many old stories in which the gods played a part (some of which may have been seen as allegorical, but others as historical truth), they also referred to these same gods as forces affecting the real world. If they were ill, they would pray to Asklepios or Apollo, and if they recovered they would dedicate offerings in thanks. If they hated someone they would call on the gods and demons to punish them, and inscribe their curses on lead tablets. Before a battle or a contest, they didn't just say "let's fight like Zeus fought the Titans"; they raised their eyes to the sky and asked Zeus for help (or Hera, or Athena, or Herakles...). If they won, they wouldn't just tell stories of their own courage, but would say this or that divinity personally assisted them. These are not traditional stories or allegories or myths; they make no reference to existing stories about the gods or their roles in legendary events. They are real-world historical or eyewitness accounts which can only be explained if we assume they are told from a perspective in which the gods are real.

For decades, folklorists followed the lead of the European folk to make a distinction between folktales, elaborate stories told as fiction, and legends, narratives told to be believed. More recently, the celebrated Swedish-Irish folklorist, Bo Almqvist (1931-2013) suggested that the definition of a legend should be that it is a story usually told to be believed, and this points to an issue in folklore studies and to one that you raise.

The problem is, the folk don't always behave themselves and stick to rigid categories. This is something that folklorists and ethnographers of more recent times have wrestled with when dealing with contemporary cultures and people. A story told in one setting may be taken to be true while in another it is taken to be fictional. Similarly, different members of an audience may regard it in different ways. This is the problem that modern scholars need to address when dealing with more recent expressions of folklore.

Given that people in a fairly universal way have told stories and made a distinction separating fiction from narratives to be believed, we can assume that the same distinction existed in ancient - and indeed in prehistoric - times. Indeed, ancient literature emerges full of folklore, but it is not always clear if the stories that are recorded in these early texts were of one or the other species. Certainly, we would expect to see expressions of both genres. It is important to point out that this is not a matter of the dichotomy that distinguishes between skeptics and the faithful. Some stories were certainly told purely for entertainment while others were believed to be true. Of those told as true, some were matter of faith while others were simply expressions of people telling about things that they assume to be true - whether they are or are not.

To determine which texts reflect an actual belief system, we can look to ancient authors, who discuss these stories as matters of belief, even if those authors themselves sometimes give voice to skepticism. After all, if I say that I do not believe in the Resurrection story, there is a built in implication that I had to say that to distinguish myself from those who do believe.

At the same time, archaeological evidence, especially in the form of temples, demonstrate that we are not looking at a body of fictional stories. There may be some level of misunderstanding the texts and what they imply about belief systems as opposed to a celebration of really good stories, but there is enough evidence to gives us good clues that in much of the material, we are looking at stories that were intended to be believed.