Given that the list contains multiple projects, and that you are inviting site visitors to choose one of them, there can be no question that the wording "Choose a project" describes the invited conduct clearly and accurately.
That doesn't mean that "Choose the project" is indefensible as wording. If I were intent on defending that wording, I would argue that "Choose the project" is simply a shortened form of a longer invitation/instruction:
Choose the project [that you'd like to examine in depth, or download, or whatever].
So it's not indefensible—and indeed, as one or more commenters above have noted, a similar truncation may be attributed to your preferred wording:
Choose a project [to examine in depth, download, or whatever].
Still, like you, I see a difference in immediate clarity between extending a deck of fanned-out cards toward someone and saying "Pick a card," and doing the same thing but saying "Pick the card."