When an acid neutralizes a base (or vice versa) it forms a #"salt"#, and water (typically we specify a water solvent when we speak of #"acid-base chemistry"#).
Consider the simplest acid-base reaction:
#NaOH(aq) + HCl(aq) rarr NaCl(aq) + H_2O(l)#
Certainly, sodium chloride, unquestionably a salt (a binary compound of a metal and a non-metal), AND water have been formed as the result of neutralization of #HO^-# and #H^+#. I write #(aq)# beside the products and reactants, because in solution these species are ionized, e.g. #HO^-#, #Cl^-#, #Na^+# etc.
The acidium species is often represented as #H_3O^+#; this again is a representation. As fas as anyone knows this is a cluster of water molecules, with an extra #H^+#, i.e. #H_5O_2^+#, #H_7O_3^+#. If we treat it as #H^+#, or #H_3O^+#, we are able to do calculations and stoichiometry.
All (aqueous) acid base chemistry can be represented by the word equation:
#"Acid + base " rarr" salt + water"#
