For a Poisson distribution with parameter $\lambda$, the probability mass function is $P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$.

Given $P(X = 1) = P(X = 3)$:

$$\frac{e^{-\lambda} \lambda^1}{1!} = \frac{e^{-\lambda} \lambda^3}{3!}$$ $$\lambda = \frac{\lambda^3}{6} \implies 6\lambda = \lambda^3 \implies \lambda^2 = 6 \implies \lambda = \sqrt{6} \approx 2.4495$$

The mode of a Poisson distribution when $\lambda$ is non-integral is the integer part of $\lambda$:

$$\text{Mode} = [\lambda] = [\sqrt{6}] = [2.4495] = 2$$