Confusion on Strategy Sets in a simultaneous move game
There is no conflict. The strategy is a function which maps decision points to actions, that is it tells the player in each decision point (to be precise, in each information set) what she should do. In a dynamic game there are usually more decisions points, so this function nature of the strategy is more obvious. 

In the simultaneous game you describe there is only one decision point (information set), players have to make a choice at the start of the game. A strategy maps an action to this one decision point (information set), so basically the strategy space is just the action space.

EDIT: 
 You write " it would be natural for me to define a strategy as a function:
$$s_i: \mathbb{R} \to \mathbb{R}."$$
There is some murkiness here, because what is the first $\mathbb{R}$ supposed to be? The strategy set of the other player? But that is not observed by player $i$, so it cannot factor into her decision. As a function the strategy would look actually look like
$$s_i: \left\{I_i\right\} \to \mathbb{R},$$
where $I_i$ is the single information set of player $i$, so $\left\{I_i\right\}$ is a singleton set. There is a bijective mapping between the vector space of these strategies and the space $\mathbb{R}$ so it does no harm to define $S_i$ as equal to $\mathbb{R}$.