Question: "But when I consider the residue field of generic point or the residue field of a point in a scheme over a non-algebraically closed field, the above analogy becomes unreasonable to me."
Let $X$ be a locally Noetherian integral scheme and let $F$ be a coherent sheaf on $X$. We may construct the torsion subsheaf $T(F)$ of $F$ in the following way: for any affine open $U⊆X$, denote $A=Γ(U,O_X)$ and $M=Γ(U,F)$. Define
$$T(F)(U)=T(M)=\{m∈M|∃a∈A \text{ non-zero divisor such that $a⋅m=0$} \}.$$
Claim: $T(F)$ is a (quasi)coherent subsheaf of $F$.
Proof: If $X$ is a locally noetherian integral scheme, there is a canonical map
$$\phi: E \rightarrow E\otimes_{\mathcal{O}_X} K_X$$
where $K_X$ is the "sheaf of quotient fields" of $X$. For any open subscheme $U:=Spec(A) \subseteq X$ it follows $K:=K_X(U)\cong K(A) \cong \mathcal{O}_{X,\eta}$
where $\eta$ is the generic point. Let $\tilde{E}:=ker(\phi)$. It follows by an exercise in Atiyah-Macdonald that if $E(U) \cong M$ that $\tilde{E}(U)\cong T(M) \subseteq M$ is the torsion sub module of $M$. Hence since $\phi$ is a map of quasi coherent sheaves it follows $T(E):=ker(\phi)$ is a quasi coherentsheaf.
Hence one usage of the generic points is to use it to define the quotient field and the sheaf $K_X$ on an integral scheme $X$. This is used to define the torsion subsheaf $T(E)\subseteq E$ for any (quasi)coherent sheaf $E$.
Note: The residue field $\kappa(x)$ of any points $x\in X$ is defined using the local ring $\mathcal{O}_{X,x}$ and its unique maximal ideal $\mathfrak{m}_x$: By definition $\kappa(x):=\mathcal{O}_{X,x}/\mathfrak{m}_x$, hence for any local section $s\in \mathcal{O}_X(U)$ and any points $x\in U$ you may consider the "value of $s$ at $x$" $s_x:=\overline{s} \in \kappa(x)$.
You "evaluate" a section $s$ at a point $x$. In algebra/geometry one seldom studies this "value" , one studies the zeros of $s$: The set of $x$ where $s_x=0$ and this is a subscheme of $U$.
Example: If $C:=\mathbb{P}^1_k$ and $s(x_0,x_1)\in H^0(D(x_0), \mathcal{O}(d))$ is a section at the open set $D(x_0)$ you may write
$$s(x_0,x_1)= \sum_i a_ix_1^{d-i}x_0^i=\sum_i a_it^ix_0^d=f(t)x_0^d$$
with $f(t)\in k[t]$ and $t:=x_1/x_0$ and it follows the zero scheme
satisfies
$$Z(s):=V(f(t))) \cong Spec(k[t]/(f(t))\subseteq D(x_0).$$
If $k$ is algebraically closed and if $f(t)=\prod_j (t-b_j)^{l_j}$ it follows
the $n$ $k$-points $b_1,..,b_n\in D(x_0)(k)$ have multiplicity $l_j$: The scheme $Z(s)$ is the set $b_j$ with multiplicities $l_j\geq 1$. One frequently studies the divisor $Z(s)$ of a section $s$ in terms of the points $b_j$ and the multiplicities $l_j$. More generally if $k$ is not algebraically closed you get a decomposition $f(t)=\prod_j p_j(t)^{l_j}$ with $p_j(t) \subseteq k[t]$ an irreducible polynomial and $l_j \geq 1$, and you may view the zero scheme $Z(s):=Spec(k[t]/(f(t))$ as the $n$ points $p_j:=(p_j(t)) \subseteq k[t]$ with multiplicities $l_j$. The residue field $\kappa(x_j):=k[t]/(p_j(t))$ will be a finite extension of $k$.
Note: You seldom study the "values of $s$ at closed points $x$". What you study is the "divisor of zeros" $Z(s)$ of $s$.
Question: "On Wikipedia entry "Residue field", it says "One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point." Can you elaborate also a bit on this?"
Answer: If $k$ is any algebraically closed field and $I:=(f_1,..,f_l) \subseteq A:=k[x_1,..,x_n]$ is a prime ideal, let $X:=Z(I)\subseteq \mathbb{A}^n_k$ be the corresponding irreducible algebraic variety. Let $g(x_1,..,x_n)\in A$ be any polynomial and view
$$g:\mathbb{A}^n_k(k) \rightarrow k$$
as a function where for any $n$-tuple $p:=(p_1,..,p_n)\in k^n$ you define $ g(p):=g(p_1,..,p_n)\in k$. You let $A(X):=A/I$ be the coordinate ring of $X$ and since $k$ is algebraically closed it follows a maximal ideal $\mathfrak{m}\subseteq A(X)$ is on the form $\mathfrak{m}:=(x_1-a_1,..,x_n-a_n)$ with $f_j(a_1,..,a_n)=0$ for all $j$. You may consider $g$ as an element in $A(X)$ and the "value" of $g$ at the "point " $\mathfrak{m}$ is the class
$$\overline{g}=g(a_1,..,a_n)\in A(X)/\mathfrak{m} \cong k.$$
This is how the polynomial $g$ is a function on the set of closed points of $X$. Hence the "natural domain" for the polynomial function $g$ is the set of closed (and non-closed) points of $X$ (and $\mathbb{A}^n_k$).
Example: If $A$ is a commutative ring and $M$ a left $A$-module and $s,t\in M$, let $I:=\{a(s-t):a\in A\}\subseteq M$ be the submodule generated by the element $s-t\in M$. By Atiyah-Macdonald Prop 3.8 it follows $I_{\mathfrak{m}}=0$ for all maximal ideals $\mathfrak{m}$ iff $I=0$. Hence there is an equality of sections $s=t$ iff $s_{\mathfrak{m}}=t_{\mathfrak{m}}$ for all maximal ideals $\mathfrak{m}$. Hence to check if the sections $s$ and $t$ give the same element in $M$, it is enough to check this at stalks at maximal ideals.
