I think you can show this using the same argument as in the continuous linear operators.
Since T is continuous, then $U=T^{-1}( (-1,1) )$ is open and contains $(0,0)$. find a c>0 small enough such that if $|x|,|y|\leq c$ then $(x,y)\in U$ and then for a general point (x,y) you have
$|T(x,y)| = |T(\frac{c x}{|x|} \frac {|x|}{c}, \frac{c y}{|y|} \frac {|y|}{c} )| = \frac {|x||y|}{c^2} |T(\frac{c x}{|x|} , \frac{c y}{|y|})| \leq \frac {|x||y|}{c^2}$
if x=0 or y=0 then T(x,y)=0 so you can use the argument above for $x,y \neq 0$.
Note: Here are two examples of partial converses which could be convenient.
Partial converse in section 27 of A Hilbert Space Problem Book by Paul R. Halmos:
Every bounded subset of a Hilbert Space is weakly bounded.
We can read in section $27$: Uniform boundedness
The celebrated principle of uniform boundedness (true for all Banach spaces) is the assertion that a pointwise bounded collection of bounded linear functionals is bounded. The assumption and the conclusion can be expressed in the terminology appropriate to a Hilbert space $\mathbf{H}$ as follows.
The assumption of pointwise boundedness for a subset $\mathbf{T}$ of $\mathbf{H}$ could also be called weak boundedness; it means that foreach $f$ in $\mathbf{H}$ there exists a positive constant $\alpha(f)$ such that $|(f,g)|\leq \alpha(f)$ for all $g$ in $\mathbf{T}$.
The desired conclusion means that there exists a positive constant $\beta$ such that $|(f,g)|\leq \beta\|f\|$ for all $f$ in $\mathbf{H}$ and all $g$ in $\mathbf{T}$; this conclusion is equivalent to $\|g\|\leq \beta$ for all $g$ in $\mathbf{T}$.
It is clear that every bounded subset of a Hilbert space is weakly bounded. The principle of uniform boundedness (for vectors in a Hilbert space) is the converse: every weakly bounded set is bounded.
And later on in section $51$: Uniform boundedness of linear transformations
... the generalization of the principle of uniform boundedness from linear functionals to linear transformations is somewhat subtler. The generalization can be formulated almost exactly the same way as the special case: a pointwise bounded collection of bounded linear transformations is uniformly bounded. The assumption of pointwise boundedness can be formulated in a weak manner and a strong one.
A set $\mathbf{Q}$ of linear transformations (from $\mathbf{H}$ into $\mathbf{K}$) is weakly bounded if for each $f$ in $\mathbf{H}$ and each $g$ in $\mathbf{K}$ there exists a positive constant $\alpha(f,g)$ such that $|(Af,g)|\leq \alpha(f,g)$ for all $A$ in $\mathbf{Q}$. The set $\mathbf{Q}$ is strongly bounded if for each $f$ in $\mathbf{H}$ there exists a positive constant $\beta(f)$ such that $\|Af\|\leq \beta(f)$ for all $A$ in $\mathbf{Q}$.
It is clear that every bounded set is strongly bounded and every strongly bounded set is weakly bounded. The principle of uniform boundedness for linear transformations is the best possible converse.
Partial converse in section $18.2$ of Mathematical Methods in Physics: Distributions, Hilbert Space Operators, and Variational Methods by Philipp Blanchard and Erwin Brüning
Every uniformly bounded family of continuous linear functionals is pointwise bounded
We can read in section $18.2$: The weak Topology:
$\mathbf{Definition\ 18.2.3}$ Let $X$ be a Banach space with norm $\|\cdot\|$ and $\{T_\alpha:\alpha\in A\}$ a family of continuous linear functionals on $X$ ($A$ an arbitrary index set). One says that this family is
- $\mathbf{pointwise\ bounded}$ if, and only if, for every $x\in X$ there is a real constant $C_x<\infty$ such that
$$\sup_{\alpha\in A}|T_\alpha(x)|\leq C_x$$
- $\mathbf{uniformly\ bounded}$ or $\mathbf{norm\ bounded}$ if, and only if
$$\sup_{\alpha \in A}\sup\{|T_\alpha(x)|:x\in X,\|x\|\leq 1\}=C< \infty.$$
Clearly, every uniformly bounded family of continuous linear functionals is pointwise bounded.
For a certain class of spaces the converse is also true and is called the principle of uniform boundedness or the uniform boundedness principle.
Maybe also interesting is the following generalisation to Hausdorff locally convex topological vector spaces stated in Appendix C: The Uniform Boundedness Principle
$\mathbf{Definition\ C.1.1}$ Suppose $(X,\mathcal{P})$ and $(Y,\mathcal{Q})$ are two Hausdorff locally convex topological vector spaces over the field $\mathbb{K}$. Denote the set of linear functions $T:X\rightarrow Y$ with $L(X,Y)$. A subset $\Lambda\subset L(X,Y)$ is called
a) $\mathbf{pointwise\ bounded}$ if, and only if, for every $x\in X$ the set $\{Tx:T\in\Lambda\}$ is bounded in $(Y,\mathcal{Q})$, i.e., for every semi-norm $q\in \mathcal{Q}$,
$$\sup\{q(Tx):T\in\Lambda\}=C_{x,q}<\infty;$$
b) $\mathbf{equi}$-$\mathbf{continuous}$ if, and only if, for every semi-norm $q\in\mathcal{Q}$ there is a semi-norm $p\in\mathcal{P}$ and a constant $C\geq 0$ such that
$$q(Tx)\leq Cp(x)\qquad\qquad\forall x\in X, \forall T\in\Lambda.$$
Obviously, the elements of an equi-continuous family of linear mappings are continuous and such a family is pointwise bounded. For an important class of spaces $(X,\mathcal{P})$ the converse holds too, i.e., a pointwise bounded family of continuous linear mappings $\Lambda\subset L(X,Y)$ is equi-continuous.
Note: The conclusion of uniform boundedness on bounded sets in the classical uniform boundedness principle for normed spaces is equivalent to the condition that the family of operators is equicontinuous with respect to the original topology of the domain spaces. See e.g. this paper by Ronglu Li and Charles Swartz.
Best Answer
Consider the family
$$\mathscr{F} = \{ f_y : y \in B_Y \}.$$
By the continuity of the $f_x$, the family is pointwise bounded,
$$\sup_{y\in B_Y} \lVert f_y(x)\rVert_Z = \lVert f_x\rVert < \infty.$$
By the uniform boundedness principle, the family is uniformly bounded, i.e.
$$\sup_{y\in B_Y} \lVert f_y\rVert = K < \infty.$$
And that says
$$\sup_{x\in B_X,\, y\in B_Y} \lVert f(x,y)\rVert_Z = K < \infty,$$
hence
$$\lVert f(x,y)\rVert_Z \leqslant K\cdot \lVert x\rVert_X\cdot \lVert y\rVert_Y.$$