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The above answers explain what is needed for the definition of a fundamental group to make sense. Let me try to answer the question from a different angle and explain what properties of the interval are needed for this notion to be reasonably well-behaved.

I believe that the key property is that, intuitively speaking, "small pieces of $I$ look just like the whole thing". More precisely, the interval can be subdivided arbitrarily finely into smaller intervals, i.e. given any open cover $\mathcal{U}$ of $I$ there is a sequence $0 = t_0 < \ldots < t_m = 1$ such that for every $i$ we have $[t_{i - 1}, t_i] \subseteq U$ for some $U \in \mathcal{U}$. In some sense this is a stronger version of the observation that gluing two intervals yields a space that is again homeomorphic to the interval. (It is interesting that the universal property of the theorem mentioned by Tom Leinster already implies the "strong version" of the subdivision property even though it is stated purely in terms of the "weak version".) This is easily proven using the Lebesgue's Lemma and is the starting point of standard techniques for calculating with fundamental groups like the path lifting property for coverings or the Van Kampen Theorem. A similar property of cubes leads to similar techniques for higher homotopy groups.

I cannot think of any other space with a property of this kind we could use in place of $I$. However, it would be interesting to see if there is some analogy between this standard approach to the fundamental group and approaches to something like Čech fundamental group (which I am unfamiliar with).

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The above answers explain what is needed for the definition of fundamental group to make sense. Let me try to answer the question from a different angle and explain what properties of the interval are needed for this notion to be reasonably well-behaved.

I believe that the key property is that, intuitively speaking, "small pieces of $I$ look just like the whole thing". More precisely, the interval can be subdivided arbitrarily finely into smaller intervals, i.e. given any open cover $\mathcal{U}$ of $I$ there is a sequence $0 = t_0 < \ldots < t_m = 1$ such that for every $i$ we have $[t_{i - 1}, t_i] \subseteq U$ for some $U \in \mathcal{U}$. In some sense this is a stronger version of the observation that gluing two intervals yields a space that is again homeomorphic to the interval. This is easily proven using the Lebesgue's Lemma and is the starting point of standard techniques for calculating with fundamental groups like the path lifting property for coverings or the Van Kampen Theorem. A similar property of cubes leads to similar techniques for higher homotopy groups.

I cannot think of any other space with a property of this kind we could use in place of $I$. However, it would be interesting to see if there is some analogy between this standard approach to the fundamental group and approaches to something like Čech fundamental group (which I am unfamiliar with).