Let $X$ be a Brauer-Severi variety over a field $k$ of characteristic $0$. In other words, suppose that $X_{\overline{k}} \cong \mathbb{P}_{\overline{k}}^n$.

I came across a statement that the map $X \times_k X \longrightarrow X$ sending an element of $X \times_k X$ to its first factor having a section (the diagonal map) implies that $X \times_k X \cong \mathbb{P}_X^n$ (the fiber product with $X$). Why is this the case? I tried to show that $X \times_k X$ has the universal property of $\mathbb{P}_X^n$, but I'm not sure why this is clear in the first place. Is there a simple way of thinking about this that I'm missing?

Some further questions:

More generally, what are other situations where maps/projections from a product of Brauer-Severi varieties having a section implies that the product is isomorphic or birational to a "similar" fiber product with projective space?

Are there suitable varieties $A$ over $k$ such that everything above works when we replace $\mathbb{P}_k^n$ with $A$?

Vague generalization: When do maps with sections induce isomorphisms with some (fiber) product?

Edit:

Given the answer below, the third question may be rephrased as follows:

When does the existence of a section imply a map is Zariski locally trivial? (e.g. Brauer-Severi varieties) How does this compare to the topological situation? (e.g. principal bundles with a global section)