To make the problem simpler, let's consider the case where the allowed values of the share of each person is discrete, e.g., integers. Equivalently, one can also imagine partitioning the "income axis" into equally-spaced intervals and approximating all values falling into a given interval by the midpoint.
Denoting the total income as $X$, the $s$-th allowed value as $x_{s}$, the total number of people as $N$, and finally, the number of people with shares of $x_{s}$ as $n_{s}$, the following conditions should be satisfied: \begin{equation} C_{1} (\{n_{s}\})\equiv\sum_{s} n_{s} - N = 0, \end{equation} and \begin{equation} C_{2} (\{n_{s}\})\equiv \sum_{s} n_{s} x_{s} - X = 0. \end{equation}
Notice that many different ways to divide the share can represent the same distribution. For example, if we consider dividing \$4 between two people, giving \$3 to Alice and \$1 to Bob and vice versa both give identical distributions. As the division is random, the distribution with the maximum number of corresponding ways to divide the share has the best chance to occur.
To obtain such a distribution, one has to maximize \begin{equation} W(\{n_{s}\}) \equiv \frac{N!}{\prod_{s} n_{s}!}, \end{equation} under the two constraints given above. The method of Lagrange multipliers is a canonical approach for this. Furthermore, one can choose to work with $\ln W$ instead of $W$ itself, as "$\ln$" is a monotone increasing function. That is, \begin{equation} \frac{\partial \ln W}{\partial n_{s}} = \lambda_{1} \frac{\partial C_{1}}{\partial n_{s}} + \lambda_{2} \frac{\partial C_{1}}{\partial n_{s}} = \lambda_{1} + \lambda_{2} x_{s}, \end{equation} where $\lambda_{1,2}$ are Lagrange multipliers. Notice that according to Stirling's formula, \begin{equation} \ln n! \approx n\ln n - n, \end{equation} leading to \begin{equation} \frac{d\ln n!}{dn} \approx \ln n. \end{equation} Thus, \begin{equation} \frac{\partial \ln W}{\partial n_{s}} \approx -\ln n_{s}. \end{equation} It then follows that \begin{equation} n_{s} \approx \exp\big(-\lambda_{1} - \lambda_{2} x_{s}\big), \end{equation} which is an exponential distribution.
That this is really a maximum, rather than a minimum or a saddle point, can be seen from the Hessian of $\ln W - \lambda_{1} C_{1} - \lambda_{2} C_{2}$. Because $C_{1,2}$ are linear in $n_{s}$, it is the same as that of $\ln W$: \begin{equation} \frac{\partial^{2} \ln W}{\partial n_{s}^{2}} = -\frac{1}{n_{s}} < 0, \end{equation} and \begin{equation} \frac{\partial^{2} \ln W}{\partial n_{s}\partial n_{r}} = 0 \quad (s\neq r). \end{equation} Hence the Hessian is concave, and what we have found is indeed a maximum.
Notice that $W(\{n_{s}\})$ is in a sense the distribution of distributions. For distributions we typically observe to be close to the most probable one, $W(\{n_{s}\})$ should be narrow enough. It is seen from the Hessian that this condition amounts to $n_{s}\gg 1$. Therefore, to actually see the exponential distribution, partitions in the income axis (corresponding to bins in OP's histogram) should be wide enough so that number of people in a partition is much greater than unity. Towards the tail, where $n_{s}$ tends to zero, this condition is always destined to fail.
Note: This is exactly how physicists understand the Boltzmann distribution in statistical mechanics.