In general, we say that something like
$$
x^2 + 3x + 2
$$
is a polynomial of degree 2. It's "nice" to write this as
$$
x^2 + 3 x^1 + 2 x^0
$$
because there's a clear pattern: the powers of $x$ decrease to zero.
But the use of $x^0$ here should be regarded as "formal" -- it's just a different way of writing the number $1$, but which fits into the nice pattern. So what you've asked about:
$$
3x^1 + 4 y^0
$$
looks like it's be undefined at $y = 0$, but in fact it's really a peculiar way of writing
$$
3x^1 + 4 \cdot 1,
$$
i.e.,
$$
3x + 4
$$
which is defined for every $xy$-point in the plane.
By asking this, you've hit on a particularly messy bit of notation, one that we (as mathematicians) tend to gloss over because we all "know what it means." To actually formally define what a polynomial is takes a bit of work, and what we teach beginning algebra students is just a way of getting them accustomed to working with polynomials without all the formal stuff...but then it runs into nasty cases like the one you've raised.
So: good for you for looking closely at what you're working with!
But: ignore this problem, and write your polynomials without any $x^0$ or $y^0$ terms, and use constants instead. Then the most general linear polynomial you need to consider is
$$
Ax + By + C
$$
where $A, B,$ and $C$ are all real numbers, and at least one of $A$ or $B$ is nonzero. (If they were both zero, then this would be a degree-0 polynomial.)
If you set this to be zero, you get
$$
Ax + By + C = 0
$$
and if $B$ is nonzero, you can rewrite that as
$$
y = -\frac{A}{B} x - \frac{C}{B}
$$
which is a line in $y = mx + b$ form. If $B$ actually IS zero, then you have
$$
Ax + C = 0\\
x = \frac{-C}{A}
$$
which is the equation of the vertical line through the point $(-\frac{C}{A}, 0)$.
