Why does the sodium element of a group $1$ and the chlorine element of group $17$ both have valency $1$?

Hint: Valency of an element is the capacity of an element to combine with another element. Valency is neither negative nor positive, it is just a number. It shows the number of electrons an element can donate or accept in order to achieve a stable configuration mostly similar to the nearest inert gas.

Complete answer:
Valence is defined as the combining capacity of an element. It shows the number of electrons the particular element can donate or accept to achieve a stable configuration.
A valency is just a number, hence it cannot be positive or negative.
Here, the valency $1$ means that the element can either donate one electron or accept one electron.
Let us consider the electron configuration of Sodium, which is given as
$Na=1{{s}^{2}}2{{s}^{2}}2{{p}^{6}}3{{s}^{1}}$
Sodium has one electron in its outermost orbit (or valence orbit) and to reach a stable configuration similar to an inert gas, it can either donate one electron or accept seven electrons. But the sodium element is very less electronegative and hence, it will donate one electron.
Hence, by saying the Sodium element has a valency of $1$ , we understand that it can donate one electron and achieve a configuration similar to the Neon element. This applies to all the group- $1$ elements.
Now, the electron configuration of chlorine is given as
$Cl=1{{s}^{2}}2{{s}^{2}}2{{p}^{6}}3{{s}^{2}}3{{p}^{5}}$
Chlorine has five electrons in its outermost orbit (or valence orbit) and to reach a stable configuration similar to inert gas, it can either donate seven electrons or accept one electron. As the Chlorine element is highly electronegative, it will accept one electron.
Hence, by saying the Chlorine element has a valency of $1$ , we understand that it can accept one electron and achieve a configuration similar to the Argon element. This applies to all the Halogen group or Group - $17\;$ elements.
Hence, even though the valency of Sodium and Chlorine is $1$, their capacity of donating or accepting electrons is the opposite.

Note:
The valency of a group or family in the periodic table is mostly equal. For example, the valency of alkali metals is $1$ , the valency of alkaline earth metals is $2$ , the valency of the halogen family is mostly $1$ , the valency of oxygen family is $2$ etc. Whereas the valency in periods before the transition elements increases from left to right till it reaches the carbon family with valency $4$ , where further it starts decreasing till it becomes zero for the noble gases.