Is cross multiplication the right term?

I always hear the term cross multiplication when it comes to equations with fractions such as $$\frac{x}{y} = \frac{a}{b}$$ where the cross multiplication is the act of multiplying $b$ and $x$, and $a$ and $y$ to get $$bx = ay.$$ What I don't get is the existence of the term at all. Isn't this just multiplication at all? This term might even get confused with the cross product.

What is the reason why cross multiplication exists even though it is just multiplication?

Cross multiplication is an informal name for the problem-solving technique, not a word with a technical definition referring to a mathematical object.

You say yourself "cross multiplication is the act of multiplying $b$ and $x$, and $a$ and $y$".

Of course, mathematically, it is "just multiplication". But humans, and in particular young students, think of operations "procedurally". The idea is to let them see the approach ("do this"), not the formal operation ("just multiplication").

[On a similar note, to students the statements $$2 \cdot 3=6$$ and $$6= 2\cdot 3$$ are not the same: They would say that the first is a multiplication, the second is factoring. And there is something to say in favour of this, on the meta-level, even though "mathematically" the two statements are completely equivalent. But a human mind does not think of a statement as only a statement, rather it naturally interprets it as part of a task, a solution to a certain problem.]

Or to put it in yet other words: Mathematically, "cross multiplication" is just a redundant term for multiplication. Pedagogically, it shows students a specific approach to a specific kind of problem.

If you connect $x$ with $b$ with a line, and then $a$ with $y$ with another line (i.e. if you connect the terms you multiply) they form a cross. That's why it's called this way.

I doubt you can confuse this with cross product because one is studied in 6th or 7th grade, the other one in university.