How to colorized any line drawed with draw (nor with plot expression)?

Here is a proposal with luadraw (which produces TikZ) using g:Dshadedpolyline():

\documentclass{standalone}%compile with lualatex only
\usepackage[svgnames]{xcolor}
\usepackage{luadraw}%https://github.com/pfradin/luadraw/
\usepackage{fourier-otf}
%Source - https://tex.stackexchange.com/q/761483
\begin{document}
\begin{luadraw}{name=test}
local g = graph:new{window={-3,3,-1,9}, size={12,12,0} }
require 'luadraw_shadedforms'
local f = function(x) return x^2 end
g:Daxes()
local L = cartesian(f,-3,3)
g:Dshadedpolyline(L,palRainbow,{values="y",width=8})
local C = {Z(-2,0.5), Z(0,-1), Z(0,7), Z(1,7.8), "b", Z(2,1), Z(2,4), Z(3,3),"b"}
-- the path(C) instruction transforms the path C into a polygonal line.
g:Dshadedpolyline(path(C),palRainbow,{values="y",width=8})
g:Show()
\end{luadraw}
\end{document}

As far as I know, this cannot be done directly with an ordinary \draw command.

The reason is that colormap in pgfplots works via point meta, and to color a line segment by segment, you need \addplot[mesh,...]; an ordinary TikZ path has no such mechanism.

So the practical approach is not to draw the curve as \draw ... .. controls ..., but to represent it as a parametric curve using \addplot and set point meta=y. Here is an example of how this can be done:

\documentclass[border=25pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    title={Color depending on $y$ coordinate},
    colormap/hot,
]

% обычный график
\addplot[
    mesh,
    ultra thick,
    domain=-3:3,
    samples=100,
    point meta=y
] {x^2};

% first cubic Bézier segment
\addplot[
    mesh,
    ultra thick,
    point meta=y,
    variable=\t,
    domain=0:1,
    samples=80,
] (
    {
      -2*(1-\t)^3 + \t^3
    },
    {
      0.5*(1-\t)^3
      - 3*(1-\t)^2*\t
      + 21*(1-\t)*\t^2
      + 7.8*\t^3
    }
);

% second cubic Bézier segment
\addplot[
    mesh,
    ultra thick,
    point meta=y,
    variable=\t,
    domain=0:1,
    samples=80,
] (
    {
      1*(1-\t)^3
      + 6*(1-\t)^2*\t
      + 6*(1-\t)*\t^2
      + 3*\t^3
    },
    {
      7.8*(1-\t)^3
      + 3*(1-\t)^2*\t
      + 12*(1-\t)*\t^2
      + 3*\t^3
    }
);

\node[anchor=east] at (axis cs:3,3.5)
  {How to colorize black line with colormap?};
\node[anchor=east] at (axis cs:3,3.0)
  {depending on $y$ coordinate};

\end{axis}
\end{tikzpicture}
\end{document}

For comparison, here is an approach using Metapost. I'm showing two different palettes: standard HSV colours and one of the colour ranges defined in Colorbrewer. In both cases I am "cheating" by drawing the line as a series of dots (but this is essentially what all the other tools are doing, they just hide it from the user).

You need to compile this with lualatex. Follow the link at the top for tutorials and reference material about Metapost.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
vardef hsv_color(expr h, s, v) =
  save chroma, hh, x, m;
  chroma = v * s;
  hh = h / 60;
  x  = chroma * (1 - abs(hh mod 2 - 1));
  m  = v - chroma;
  if     hh < 1: (chroma,x,0)+(m,m,m)
  elseif hh < 2: (x,chroma,0)+(m,m,m)
  elseif hh < 3: (0,chroma,x)+(m,m,m)
  elseif hh < 4: (0,x,chroma)+(m,m,m)
  elseif hh < 5: (x,0,chroma)+(m,m,m)
  else:          (chroma,0,x)+(m,m,m)
  fi
enddef;
beginfig(1);
  numeric u, v;
  u = 2v = 25;

  path xx, yy;
  xx = (left--right) scaled 3u;
  yy = origin -- up scaled 9v;

  numeric s; s = 1/8;
  path ff;
  ff = (-3,9) for x = s-3 step s until 3: -- (x, x*x) endfor;
  ff := ff xscaled u yscaled v;

  for t = 0 step 1/8 until arclength ff:
    pair w; w = point arctime t of ff of ff;
    draw w withcolor hsv_color(30(9 - ypart w / v),2,1); 
  endfor

  path gg;
  gg = (-2,0.5) .. controls (0,-1) and (0,7) .. (1,7.8) .. controls (2,1) and (2,4) .. (3,3);
  gg := gg xscaled u yscaled v;

  input colorbrewer-rgb
  for t = 0 step 1/8 until arclength gg:
    pair w; w = point arctime t of gg of gg;
    numeric p, q; 
    p = 8 - ypart w / v;
    q = floor p; 
    draw w withcolor (p-q)[PiYG[9][q+1], PiYG[9][q+2]];
  endfor

  ahangle := 20;
  drawoptions(withcolor 1/2);
    drawdblarrow xx scaled 1.2;
    drawarrow yy scaled 1.1;
    for x = -3,-2,-1,1,2,3: 
      draw (down--up) scaled 2 shifted (x*u,0);
      label.bot("$" & decimal x & "$", (x*u, -2));
    endfor
    for y = 2, 4, 6, 8:
      draw (left--right) scaled 2 shifted (0, y*v);
      label.lft("$" & decimal y & "$", (-2, y*v));
    endfor
  drawoptions();
endfig;
\end{mplibcode}
\end{document}

A draw can only have one color. As you show, you can use the option mesh to split the path into multiple colored segments. -this is the "correct" methode. If you really insist on a draw instead of a plot, it can actually be done with a tikzfadingfrompicture although I can not recommend this methode:

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{fadings}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepgfplotslibrary{patchplots}

\begin{tikzfadingfrompicture}[name=myfading]
\begin{axis}[axis lines=middle, draw=none, xmin=-3, xmax=3, ymin=0, ymax=9]
\draw[transparent!0, thick, xshift=0.5, rounded corners] (-2,0.5) .. controls (0,-1) and (0,7) .. (1,7.8) .. controls (2,1) and (2,4) .. (3,3);
\end{axis}
\path (0,0) circle[radius=10]; %encompassing circle for alignment
\end{tikzfadingfrompicture}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
axis lines=middle, xmin=-3, xmax=3, ymin=0, ymax=9,
colormap/hot,
]
\addplot[mesh, thick, domain=-3:3, samples=100, point meta=y] {x^2};
\path[scope fading=myfading, fit fading=false] (0,0);
\addplot[patch, shader=interp, patch type=rectangle, patch refines=3] coordinates{(-3,0) (-3,9) (3,9) (3,0)};
\end{axis}
\end{tikzpicture}
\end{document} 

Here is another apporach, using masking group technique provided by luamplib package. The code is based on Truston's answer above, but the lines are real lines, not thousands of dots with different colors. As for colors, I chose the closest colors in my eyes, as I do not have information about colormaps.

Of course, it needs to compile with lualatex.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\mpfig*
  numeric u, v;
  u = 2v = 25;

  path xx, yy;
  xx = (left--right) scaled 3u;
  yy = origin -- up scaled 9v;

  path ff;
  ff = origin .. controls (1/3, 0) and (2/3, 1/3) .. (1,1);
  ff := reverse ff reflectedabout (down, up) & ff;
  ff := ff xscaled 3u yscaled 9v;

  path gg;
  gg = (-2,0.5) .. controls (0,-1) and (0,7) .. (1,7.8) .. controls (2,1) and (2,4) .. (3,3);
  gg := gg xscaled u yscaled v;

  picture bgbox;
  bgbox = image(
    save bboxmargin; bboxmargin := 1/2;
    fill bbox ff
      withshadingmethod "linear"
      withshadingvector (0,3)
      withshadingstep (
        withshadingfraction 1/3
        withshadingcolors (blue, (1,1,0))
      )
      withshadingstep (
        withshadingfraction 3/3
        withshadingcolors ((1,1,0), red)
      )
  );
\endmpfig

\mplibgroup{linesmask}[asgroup="masking"]
  \mpfig
    pickup pencircle scaled 1;
    draw ff withcolor 1;
    draw gg withcolor 1;
  \endmpfig
\endmplibgroup

\mpfig
  draw bgbox withmaskinggroup "linesmask";

  ahangle := 20;
  drawoptions(withcolor 1/2);
  drawdblarrow xx scaled 1.2;
  drawarrow yy scaled 1.1;
  for x = -3,-2,-1,1,2,3:
    draw (down--up) scaled 2 shifted (x*u,0);
    label.bot("$" & decimal x & "$", (x*u, -2));
  endfor
  for y = 2, 4, 6, 8:
    draw (left--right) scaled 2 shifted (0, y*v);
    label.lft("$" & decimal y & "$", (-2, y*v));
  endfor
  drawoptions();
\endmpfig
\end{document}