Is it possible to map from $\mathbb{F}[x,y]$ to $\mathbb{F}[u]$ where each monomial $x^ay^b = u^{aw+b}$ for appropriate choice of $w$?
In fact this mapping $\phi(x^ay^b) = u^{aw+b}$ seems to be multiplicative $\phi(z_1)\phi(z_2) = \phi(z_1z_2)$.
Is it a ring homomorphism? What is it called in literature?
The family $\{x^ny^m\ \mid\ n,m\in\mathbb N_0\}$ is a basis for $\mathbb F[x,y]$ as $\mathbb F$-vector space. Hence any choice of a family of polynomials $F_{nm}(u)\in\mathbb F[u]$ will naturally extend to a $\mathbb F$-linear map $F\colon\mathbb F[x,y]\to\mathbb F[u]$. If additionally
$$F((x^{n_1}y^{m_1})(x^{n_2}y^{m_2}))=F(x^{n_1}y^{m_1})F(x^{n_2}y^{m_2})$$
this map will be a ring homomorphism. You can check that this suffices by explicitely computing the product of two polynomials in $\mathbb F[x,y]$ using the $\mathbb F$-linearity of $F$.
In your case $F_{nm}(u)=u^{nw+m}$ and
\begin{align*} F((x^{n_1}y^{m_1})(x^{n_2}y^{m_2})) &=F(x^{n_1+n_2}y^{m_1+m_2}) \\ &=u^{(n_1+n_2)w+(m_1+_2)} \\ &=u^{n_1w+m_1}u^{n_2w+m_2} \\ &=F(x^{n_1}y^{m_1})F(x^{n_2}y^{m_2})\ . \end{align*}
So $F$ is a ring homomorphism as desired. For the map reuns proposed you can choose
$$F_{nm}(u)=g(u)^nh(u)^m$$
and multiplicativity is straigtforward as well. For $g(u)=u^w$ and $h(u)=u$ this becomes your map again.