Aligning object in a commutative diagram

Quick and dirty.

\documentclass[a4paper]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usepackage{faktor}
\begin{document}
 I'd like the first line of the following diagram to be aligned like the first line of the second diagram, with the parentheses centered:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \raisebox{-0.8em}{$\faktor{\begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix}}{\cong_b}$} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \Bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\Bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
 Second diagram:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \Bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\Bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
\end{document}

If you do not want to guess this value but compute it, you can do so as follows (with some tricks from this answer):

\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usetikzlibrary{calc}
\usepackage{faktor}
\begin{document}
 I'd like the first line of the following diagram to be aligned like the first line of the second diagram, with the parentheses centered:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm},execute at end picture={%
  \path let \p1=($(\tikzcdmatrixname-1-2.north)-(\tikzcdmatrixname-1-1.north)$)
  in \pgfextra{\typeout{required\space raise=\y1}};}]
   \raisebox{-7.25006pt}{$\faktor{\begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix}}{\cong_b}$} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \Bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\Bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
 Second diagram:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \Bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\Bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
\end{document}

If you mean the vertical placement, then the culprit is \faktor.¹

\documentclass[a4paper]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usepackage{faktor}

\begin{document}
 I'd like the first line of the following diagram to be aligned like the first line of the second diagram, with the parentheses centered:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix}_{\raisebox{\depth}[0pt][0pt]{$\big/{\cong_b}$}} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
Alternative, more in line with conventional mathematical typesetting
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix}\Big/{\cong_b} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
If you insist in using faktor
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{gathered}\faktor{\begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix}}{{\cong_b}}\end{gathered} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
 Second diagram:
 \[
  \begin{tikzcd}[row sep={0cm}, column sep={2cm}]
   \begin{Bmatrix}
    \text{Rivestimenti connessi} \\
    \text{per archi di $(X,x_0)$}
   \end{Bmatrix} \arrow[r, "\phi", leftrightarrow] &
   \begin{Bmatrix}
    \text{Sottogruppi di} \\
    \text{$\pi_1(X,x_0)$}
   \end{Bmatrix} \\
   \bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\bigr) \arrow[r, mapsto] & p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
  \end{tikzcd}
 \]
\end{document}

Footnote

¹ I find the output of faktor ugly and unnecessary: I've rarely seen such constructions in mainstream mathematical publications.

Using only the mathmode:

\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\makeatletter
\def\mapstofill@{\arrowfill@{\mapstochar\relbar}\relbar\rightarrow}
\newcommand*\xmapsto[2][]{\ext@arrow 0000\mapstofill@{#1}{#2}}
\newcommand\xlongleftrightarrow[2][]{\ext@arrow 0055{\longleftrightarrowfill@}{#1}{#2}}
\def\longleftrightarrowfill@{\arrowfill@\leftarrow\relbar\rightarrow}
\makeatother
\newcommand\TAB[1]{\tabular{@{}l@{}}#1\endtabular}

\begin{document}
I'd like the first line of the following diagram to be aligned like the first line of the second diagram, 
with the parentheses centered:
\begin{flalign*}
   \left\{\TAB{Rivestimenti connessi\\per archi di $(X,x_0)$}\right\}_{\slash\cong_b} 
 \xlongleftrightarrow{\makebox[2cm]{$\phi$}} 
  \TAB{Sottogruppi di\\ $\pi_1(X,x_0)$}    \\
   \Bigl(p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0)\Bigr) \xmapsto{\makebox[3cm]{}}
  p_*\bigl(\pi_1(\tilde{X},\tilde{x}_0)\bigr)
\end{flalign*}

\end{document}