Move label of an angle in Tikz

The text pic text has the option text pic options. Here, simply move the text up to the left to avoid overlapping on the side z_1of the angle.

pic text options={shift={(-3pt,3pt)}}

I took the liberty of colouring the different angles, which in my humble opinion makes the figure easier to read.

\documentclass{article}
\usepackage{tkz-euclide}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{subcaption}

\usetikzlibrary{calc,patterns,angles,quotes}

\begin{document}

\begin{tikzpicture}[scale=1]
 \begin{scope}[thick,font=\scriptsize]
    \draw [->] (-.5,0) -- (5.5,0) node [above left]  {$\operatorname{Re} z$};
    \draw [->] (0,-.5) -- (0,4) node [below right] {$\operatorname{Im} z$};
  \end{scope}
  \coordinate (o) at (0,0);
  \coordinate (z1) at (4.511,1.642);
  \coordinate (z2) at (2.143,2.553);
  \coordinate (pr) at (1.368,3.759);
  \coordinate (x) at (1,0);
  \coordinate (y) at (0,1);
  \draw (o) -- node[above,shift={(.4cm,.1cm)}] {$r_1$} (z1) node [right] {$z_1$};
  \draw (o) -- node[above,yshift=.2cm] {$r_2$} (z2) node [right] {$z_2$};
  \draw (o) -- node[above,yshift=1cm] {$r_1{\cdot}r_2$} (pr) node [right] {$z_1\cdot z_2$};
  \draw pic[draw,red, "$\theta_1$", ->, angle eccentricity=1.2,angle radius = 1cm] {angle = x--o--z1}; 
  \draw pic[draw,blue, "$\theta_2$", ->, angle eccentricity=1.2,angle radius = 1.5cm,pic text options={shift={(-3pt,3pt)}}] {angle = x--o--z2}; 
  \draw pic[draw,violet, "$\theta_1{+}\theta_2$", ->, angle eccentricity=1.2,angle radius = 2.5cm] {angle = x--o--pr}; 
\end{tikzpicture}

\end{document}

In this case I think that we can add options to the angle label by putting them after the quotted text like that : \draw pic[draw, "$\theta_2$"{option1,option2,...}, ->, angle eccentricity=1.2,angle radius = 1.5cm] {angle = x--o--z2};

the code

\documentclass{article}
\usepackage{tkz-euclide}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usetkzobj{all}
\usepackage{subcaption}

\usetkzobj{all}
\usetikzlibrary{calc,patterns,angles,quotes}

\begin{document}

\begin{tikzpicture}[scale=1]
 \begin{scope}[thick,font=\scriptsize]
    \draw [->] (-.5,0) -- (5.5,0) node [above left]  {$\operatorname{Re} z$};
    \draw [->] (0,-.5) -- (0,4) node [below right] {$\operatorname{Im} z$};
  \end{scope}
  \coordinate (o) at (0,0);
  \coordinate (z1) at (4.511,1.642);
  \coordinate (z2) at (2.143,2.553);
  \coordinate (pr) at (1.368,3.759);
  \coordinate (x) at (1,0);
  \coordinate (y) at (0,1);
  \draw (o) -- node[above,shift={(.4cm,.1cm)}] {$r_1$} (z1) node [right] {$z_1$};
  \draw (o) -- node[above,yshift=.2cm] {$r_2$} (z2) node [right] {$z_2$};
  \draw (o) -- node[above,yshift=1cm] {$r_1{\cdot}r_2$} (pr) node [right] {$z_1\cdot z_2$};
  \draw pic[draw, "$\theta_1$", ->, angle eccentricity=1.2,angle radius = 1cm] {angle = x--o--z1}; 
  \draw pic[ draw,->,blue, "$\theta_2$"{shift=(80:0.35),inner sep=1pt, circle,draw},angle eccentricity=1.1,angle radius = 1.5cm] {angle = x--o--z2}; 
  \draw pic[draw, "$\theta_1{+}\theta_2$", ->, angle eccentricity=1.2,angle radius = 2.5cm] {angle = x--o--pr}; 
\end{tikzpicture}

\end{document}

Let us move labels by hand.

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[scale=2,thick]
\draw[->] (-.5,0)--(2.5,0) node[below]{\rm{Re}$z$};
\draw[->] (0,-.5)--(0,3) node[left]{\rm{Im}$z$};
\def\aone{20} \def\atwo{45}
\def\rone{2}  \def\rtwo{1.6}
\draw 
(0,0)--(\aone:\rone) 
node[pos=.8,below]{$r_1$} node[right]{$z_1$} 
(0,0)--(\atwo:\rtwo) 
node[pos=.65,left]{$r_2$} node[above]{$z_2$}
(0,0)--(\aone+\atwo:\rone*\rtwo) 
node[pos=.65,left]{$r_1\cdot r_2$} node[right]{$z_1\cdot z_2$};
\draw[->,red] (0:.5) arc(0:\aone:.5) 
node[pos=.55,right]{$\theta_1$};
\draw[->,blue] (0:.8) arc(0:\atwo:.8) 
node[pos=.75,right]{$\theta_2$};
\draw[->,magenta] (0:1.2) arc(0:\aone+\atwo:1.2)
node[pos=.55,right]{$\theta_1+\theta_2$};
\end{tikzpicture}
\end{document}