Consider two probability spaces, e.g.,
Probability space 1:
the sample space $\Omega_1\equiv[0,1]$
the $\sigma$-algebra equal to the Borel $\sigma$-algebra on $[0,1]$, $\mathcal{B}([0,1])$
the measure $\mathbb{P}_1:\mathcal{B}([0,1])\rightarrow [0,1]$
Probability space 2:
the sample space $\Omega_2\equiv[0,2]$
the $\sigma$-algebra equal to the Borel $\sigma$-algebra on $[0,2]$, $\mathcal{B}([0,2])$
the measure $\mathbb{P}_2:\mathcal{B}([0,2])\rightarrow [0,2]$
What can I say about the $\sigma$-algebra $\mathcal{F}_3$ and the probability measure $\mathbb{P}_3$ of the cartesian product $\Omega_3\equiv\Omega_1\times \Omega_2$? Is it obvious that
$$
\mathbb{P}_3\equiv \mathbb{P}_1\times \mathbb{P}_2
$$
and
$$
\mathcal{F}_3\equiv \mathcal{B}([0,1])\times \mathcal{B}([0,2])
$$
or it is just one of the million possible assumptions we could make
?
When someone refers to the product $\Omega_3=\Omega_1\times \Omega_2$ as a probability space, that always means they are taking the product $\sigma$-algebra and product measure unless the context makes it clear they mean something else.
That said, it's certainly not true that this is the only possible probability space structure on the set $\Omega_1\times\Omega_2$. It's just the natural one that comes from the given probability space structures on $\Omega_1$ and $\Omega_2$, and so by convention this is the "default" structure to use if you don't say otherwise.