To answer my own question, the Lorenz curve is indeed a PP-plot, but in a different sense than I originally imagined. In particular, we can use various theoretical distributions to model the income distribution, but the resulting plot is always a comparison of $\mathrm{U}(0,1)$ CDF against a distribution with support $[0,1]$. In case the wealth is equally distributed, this distribution is also uniform, meaning the PP plot is a straight line.
First of all, it may be more productive to think about Lorenz curves in a non-probabilistic sense. Levine (1970) gives a formal description of Lorenz curves in the original sense as a way to quantify the income inequality, which I'm summarising below:
Let $t\in[0,\infty)$ denote the income, $f(t)$ the income density over people, i.e. $f(t)\mathrm{d} t$ approximately equals to the
number of people earning income in the range $(t,t+\mathrm{d} t)$. To reiterate, the integral of $f(t)$ over its domain is not equal to 1, so it's not a PDF, rather it is the total number of people:
$$
f_{0}=\int_{0}^{\infty}f(t)\mathrm{d} t.
$$
The total income earned by the people who earn income in the range
$(t,t+\mathrm{d} t)$ is $t\,f(t)\mathrm{d} t$. Then the total income of the population
is $\int_{0}^{\infty}tf(t)\mathrm{d} t$, and the average income per person
is
$$
\mu=\frac{1}{f_{0}}\int_{0}^{\infty}tf(t)\mathrm{d} t.
$$
Then the following two quantities are compared:
- The fraction of the population earning $z$ or less:
$$
F(z)=\frac{1}{f_{0}}\int_{0}^{z}f(t)\mathrm{d} t
$$
- The fraction of the total income earned by the people with income
$z$ or less:
$$
G(z)=\frac{\int_{0}^{z}tf(t)\mathrm{d} t}{\int_{0}^{\infty}tf(t)\mathrm{d} t}=\frac{1}{f_{0}}\frac{1}{\mu}\int_{0}^{z}tf(t)\mathrm{d} t.
$$
The Lorenz curve is a parametric curve given by
$$
\left\{ \begin{aligned}x & =F(z)\\
y & =G(z).
\end{aligned}
\right.
$$
We could've used normalised density
$\tilde{f}(t)=f(t)/f_{0}$ with $\int_{0}^{\infty}\tilde{f}(t)\mathrm{d} t=1$.
I guess in this case it is correct to say that $f(t)$ is the density of the "income" random variable. Below $f(t)$ has unit integral over the domain.
The Lorenz curve equations can be implicitised using a suitable change of variable (and the inverse function rule), i.e. let $F(z)=p$; then $z=F^{-1}(p)$ and
$$
G(z)=L(p)=\frac{1}{\mu}\int_{0}^{p}F^{-1}(u)\,\mathrm{d} u.
$$
The equation above gives a reason to interpret the Lorenz curve as a PP-plot of $F(z)=p=\mathrm{cdf}_{\mathrm{U}(0,1)}(p)$ against a variable $Y$ with density $F^{-1}(p)/\mu$. Indeed, the curve is a graph in $[0,1]\times [0,1]$. By construction, $\mathrm{supp}Y = [0,1]$. Moreover, since $z \mathrm{d}u$ has the meaning of "$\mathrm{d}u$ proportion of the population earns $z$", $$\int_{0}^{1}F^{-1}(u)\mathrm{d} u=\int_{0}^{1}z(u)\mathrm{d} u=\mu$$ is the average income population. The wealth is equally distributed if the fraction of the population
earning $z$ or less equals the fraction of the total income earned
by this group. In other words, $F(z)\equiv G(z)$, or, equivalently,
$L(p)=p,$ and the Lorenz curve is a straight line. This implies the density $F^{-1}(p)/\mu$ is constant. In fact, because $\mathrm{supp}Y = [0,1]$, $F^{-1}(u)/\mu=1$ and $z\equiv\mu$ (the only possible income for everybody is $\mu$). Implying in the "ideal" case $Y \sim \mathrm{U}(0,1)$.
I think this makes some sense. Below is another way of looking at the "ideal" case.
Interestingly, equal wealth distribution is not described by the uniform distribution of income. For simplicity, suppose $f(t)=1/\theta$, $t\in[0,\theta]$ (a uniform density with $a=0$). Then
\begin{align*}
F(z) & =\int_{0}^{z}f(t)\mathrm{d} t=\int_{0}^{z}1/\theta\,\mathrm{d} t=\frac{z}{\theta},\quad z\in[0,\theta]\\
\mu & =\int_{0}^{\infty}tf(t)\mathrm{d} t=\frac{1}{\theta}\int_{0}^{\theta}t\mathrm{d} t=\frac{\theta}{2}.\\
G(z) & =\frac{1}{\mu}\int_{0}^{z}tf(t)\mathrm{d} t=\frac{2}{\theta}\frac{1}{\theta}\int_{0}^{z}t\mathrm{d} t=\frac{z^{2}}{\theta^{2}}=F^{2}(z)\\
F^{-1}(p) & =\theta p\\
L(p) & =\frac{1}{\mu}\int_{0}^{p}F^{-1}(u)\mathrm{d} u=\frac{2}{\theta}\frac{\theta}{2}p^{2}=p^{2}.
\end{align*}
In fact the case of equal wealth distribution corresponds to a Dirac's delta distribution. In other words,
$$
F(z)=G(z)\ \forall z\in[0,\infty)\iff\int_{0}^{z}f(t)\mathrm{d} t=\frac{1}{\mu}\int_{0}^{z}tf(t)\mathrm{d} t\iff f(t)=\frac{1}{\mu}tf(t)\ \forall t\in0:z,
$$
which implies that $f(t)$ is the Dirac's delta
$$
\delta_{\mu}(t)=\begin{cases}
0 & t\neq\mu\\
1 & t=\mu.
\end{cases}
$$
Then
$$
F(z)=\begin{cases}
0 & z<\mu\\
1 & z\geq\mu
\end{cases},\quad G(z)=\begin{cases}
0 & z<\mu\\
\mu/\mu=1 & z\geq\mu.
\end{cases}
$$
Therefore, the Lorenz curve is $\{(0, 0), (1, 1)\}$.
Levine (1970): Levine, Singer -- The mathematical relation between the income density function and the measurement of income inequality (1970)
