Alignment and centering of pgfplots

Put all the axis environments in the same tikzpicture, and position the axes relative to the other using the at key, some shifting, and the various origin anchors. For example, for the FourierA axis, use

at={(RectA.right of origin)},
xshift=1cm,
anchor=left of origin,

Those anchors are described in section 4.19 Alignment options of the pgfplots manual. Basically, left of origin is on the left side of the axis, at the same height as the origin. Similarly for right of/above/below.

Hence, the code above will place the left of origin anchor of the FourierA axis on the right of origin anchor of RectA, but shifted 1cm right. Modify 1cm to whatever you prefer.

To place the arrows, make a path between the two axes, and add the node to that path, e.g.

path (RectA) -- node{$xLongleftrightarrow{mathcal{F}}$} (FourierA);

documentclass{beamer}
usepackage[utf8]{inputenc}

% for plotting mathematical functions
usepackage{pgfplots}
usepackage{makecell}
usepackage{array}
usepackage{amsmath}
usepgfplotslibrary{groupplots}

% For text positioning
usepackage{textpos}
usepackage{graphics}

% boldface math symbols
usepackage{bm}
usepackage{mathtools}
usepackage{extarrows}

% For drawing block diagrams, plotting, etc
usetikzlibrary{shapes, arrows, arrows.meta, positioning, calc, quotes, backgrounds,intersections, fit, matrix}
pgfplotsset{compat=1.16}


begin{document}
begin{frame}
begin{center}
begin{tikzpicture}[scale=1.0, every node/.style={scale=1.0}]
    begin{axis}[name=RectA, 
     xmin=-1.25, xmax = 2.5, 
        ymin = 0, ymax = 1.25,
        axis x line=middle, height=4.0cm, width=5.0cm, 
        x axis line style={thick},
        axis y line=middle, 
        y axis line style={thick},
        %title = {Square wave},
        ytick={1},
        yticklabels={$a$},
        xtick={-1.0,1.0},
        xticklabels={scriptsize $-T//2$,$T//2$},
        tick style={draw=none}, 
        y tick label style={font=small, xshift={-0.2cm},yshift={0.2cm}},
        x tick label style={font=scriptsize},
        xlabel={$scriptstyle t$}, 
        xlabel style={xshift=0.3cm, yshift=-0.1cm}, 
        ylabel={$x(t)$}, 
        ylabel style={font=small,yshift=0.2cm},
        ]
        addplot+[thick,mark=none,const plot]
        coordinates
        {(-1.25,0) (-1.0,1) (1.0,1) (1.0,0) (3.75,0)};
    end{axis}%
    
    begin{axis}[
    at={(RectA.right of origin)},
    xshift=1cm,
    anchor=left of origin,
    name=FourierA, 
    xmin=-6*pi, xmax = 6*pi, 
    ymin = -0.25, ymax = 1.2,
    axis x line=middle, 
    height=4.5cm, width=6.0cm, 
    x axis line style={thick},  
    axis y line=middle, 
    y axis line style={thick},
    ytick={1},
    yticklabels={$scriptstyle aT$},
    xticklabels={draw=none},
    xlabel={$scriptstyle Omega$}, 
    xlabel style={xshift=0.3cm,yshift=-0.1cm}, ylabel={$scriptstyle X(jOmega)$}, 
    ylabel style={xshift=0.0cm},
    ]
        addplot [red, dashed, domain = -6*pi:6*pi, samples = 100] {sin(2*x*180/pi)/(2*x) };
    end{axis}%
    
begin{axis}[name=RectB,
      at={(RectA.below origin)},
      yshift=-1cm,
      anchor=above origin,
       xmin=-1.25, xmax = 2.5, 
        ymin = 0, ymax = 1.25,
        axis x line=middle, height=4.0cm, width=5.0cm, 
        x axis line style={thick},
        axis y line=middle, 
        y axis line style={thick},
        %title = {Square wave},
        ytick={1},
        yticklabels={$a$},
        xtick={-0.25,0.25},
        xticklabels={scriptsize $-tfrac{T}{8}$,$tfrac{T}{8}$},
        tick style={draw=none}, 
        y tick label style={font=small, xshift={-0.2cm},yshift={0.2cm}},
        x tick label style={font=scriptsize},
        xlabel={$scriptstyle t$}, 
        xlabel style={xshift=0.3cm, yshift=-0.1cm}, 
        ylabel={$x(t)$}, 
        ylabel style={font=small,yshift=0.2cm},
        ]
        addplot+[thick,mark=none,const plot]
        coordinates
        {(-1.25,0) (-0.25,0) (-0.25,1) (0.25,1) (0.25,0) (3.75,0)};
    end{axis}

begin{axis}[name=FourierB,
    at={(RectB.right of origin)},
    xshift=1cm,
    anchor=left of origin,
    xmin=-6*pi, xmax = 6*pi, 
    ymin = -0.25, ymax = 1.2,
    axis x line=middle, 
    height=4.5cm, width=6.0cm, 
    x axis line style={thick},  
    axis y line=middle, 
    y axis line style={thick},
    ytick={0.25},
    yticklabels={$scriptstyle  aT//4$},
    xticklabels={draw=none},
    xlabel={$scriptstyle Omega$}, 
    xlabel style={xshift=0.3cm,yshift=-0.1cm}, ylabel={$scriptstyle X(jOmega)$}, 
    ylabel style={xshift=0.0cm},
    ]
        addplot [red, dashed, domain = -6*pi:6*pi, samples = 100] {0.25*sin(0.5*x*180/pi)/(0.5*x) };
    end{axis}
    
path (RectA) -- node{$xLongleftrightarrow{mathcal{F}}$} (FourierA);
path (RectB) -- node{$xLongleftrightarrow{mathcal{F}}$} (FourierB);
end{tikzpicture}
end{center}
end{frame}
end{document}