It's up to the DM, but the system is written assuming a single god, with one exception.
After reading the character creation section and the piety section, it seems abundantly clear that the rules are assuming you only follow a single god, but it is never explicitly stated. I can offer a few examples, though there are more.
From the Character Creation section.
Many characters’ ideals come directly from their service to a god. Each god’s description in chapter 2 includes a table you can use to determine your hero’s ideal if your character serves that god, instead of (or in addition to) an ideal derived from your background.
On supernatural gifts:
A character in Theros begins with one supernatural gift chosen from those in this section. Work with the DM to decide where your character’s gift came from. Is it tied to the god you serve?
From the God of Theros section.
On the difference between heroes and mere mortals:
It’s far more common for a hero to be devoted to an individual god than it is for an ordinary mortal.
On starting out:
When you choose a god to worship as a beginning character, your piety score related to that god is 1. Your piety score increases by 1 when you do something to advance the god’s interests or behave in accordance with the god’s ideals.
On piety score:
As a general rule, you can expect to increase your piety by 1 during most sessions of play, assuming that you are following your god’s tenets.
Notably, piety score is always referred to singularly.
The guidance given for gods working together assumes each champion follows one god.
In the section on building Theros adventures, Champions and Quests, the book gives some world-building guidance on how to integrate a party into the world and relate them to the diverse pantheon of Theros:
Serving Two or More Masters
Often, each of the characters in an adventuring party is the champion of a different god—or no god at all. What brings the characters together? How do you motivate the whole party? Consider these options:
Divine Coalition. The characters represent a coalition of gods who have joined forces against a common foe (or cabal of enemies). For example, you might build a campaign around the idea that Erebos, Mogis, and Pharika are conspiring to unleash slaughter and plague on the mortal realm. Perhaps heroic champions of Ephara, Heliod, Karametra, Nylea, and other gods unite to protect the world from this threat.
Friendly Cooperation. If Thassa’s champion helps Heliod’s champion today, perhaps Heliod’s champion will help Thassa’s champion tomorrow. As long as the aims of gods aren’t in opposition, those gods take no issue with their champions helping each other.
This seems to indicate that when gods are working together or have somewhat aligned goals, their champions work together, rather than the gods going all in to share a single champion. Though, this idea has one exception, noted in the next section.
The Oracle may follow multiple gods.
From the description of the Oracle supernatural gift:
The gods seek mortal oracles to act as their agents. As a result, most oracles devote themselves to the service of a single god and learn to ignore the voices of all others. Occasionally, two gods agree to share the services of an oracle. Oracles who try to remain independent often find themselves pursued by the agents of evil gods who would bind them to the god’s service, if necessary.
With this gift, the rules explicitly mention following two gods, but this gift precludes the use of the traditional piety system:
If you choose the Oracle supernatural gift, you gain different rewards for your piety score, instead of the ones normally granted by your god.
The only section that explicitly mentions serving more than a single god has an entirely unique piety system associated with it.
Incidentally, the flavor text quoted in the question for Eutropia, the Twice Favored seems to describe the work of an oracle:
Nylea speaks to her in the windblown leaves; Thassa in the rushing tides.
Thanks to user Ben Barden for pointing this out in the comments.
The message here is clear: the intent is that you only benefit from a single set of piety derived blessings at a time.
On the other hand, a DM may allow a player to play a character who follows two gods and has to juggle their responsibilities to each. The issue with this is that the divine blessings given by the piety system are quite good. A wise DM would consider the balance implications of permitting a character to double-dip, so to speak, into these blessings. Notably, the +50 piety blessings for every god is a +2 to an ability score and its max - equivalent to an epic boon. Allowing one player to double dip here is so significant a benefit, mechanically minded players will almost certainly find a way to shoehorn it into their back story.