Double up some diamonds

SOGL V0.12, 24 bytes

ā.∫ā;∫ \*+}ø┼↔±╬¡;øΚ┼}╬³

Try it Here!

Explanation:

ā                         push an empty array (the main canvas)
 .∫                  }    iterate over the input, pushing 1-indexed iteration
   ā;                       push an empty array below the iteration
     ∫    }                 iterate over the iteration counter
       \*                     push current iteration amount of slashes
         +                    append those to the 2nd array
           ø┼               append nothing (so it'd space the array to a square)
             ↔±             reverse horizontally (swapping slashes)
               έ           quad-palindromize with 0 overlap and swapping characters as required
                 ;          get the canvas ontop
                  øΚ        prepend to it an empty line (so the now bigger romb would be one up)
                    ┼       append horizontally the canvas to the current romb
                      ╬³  palindromize horizontally with no overlap and swapping characters

Charcoal, 30 27 bytes

Ｆ⁺¹Ｎ«Ｆι«Ｆ⁴«↙⁻ικ↑⟲Ｔ»←»Ｍι←»‖Ｍ

Try it online! Link is to verbose version of code. Explanation: Charcoal's drawing primitives can't quite draw a diamond, because the diagonal movements stay on squares of the same parity. Edit: The new solution is to draw one side of a diamond and then rotate the entire canvas ready to draw the next side, allowing a diamond to be drawn in a loop. This loop is then contained in a loop to draw all the inner diamonds for each diamond. The outermost loop draws all diamonds adjacent to each other. Finally the image is mirrored.

Note that Charcoal has since been extended and another byte could be saved by using Increment.

05AB1E, 47 43 41 35 34 33 32 bytes

'/×ηηvy∞.C.Bø€∞¹NαGð.ø}})øíJ.B»∞

Try it online!

(-4 bytes thanks to @Emigna who suggested 3 improvements)

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This explanation was for the earlier version, there have been a few iterations since then.

>                                          # [2]
 '/×                                       # ['//']
    η                                      # ['/','//']
     €η                                    # [['/'], ['/', '//']]
       vy                    }             # {For each element...}
         ∞                                 # Mirror horizontally.
          ¶¡                               # Split mirror on newlines.
            N£                             # Shave each diamond down a layer.
              .C                           # Horizontal center.
                .B                         # Pad for the transpose.
                  ø                        # Transpose.
                   €∞                      # Mirror each (vertically).
                     ¹NαFð.ø}              # Pad each list for transpose (verticaly).
                              )            # Wrap back to list...
                               €.B         # Pad each horizontally.
                                  ¦        # Remove the random numbers?
                                   ø       # Back to horizontal...
                                    €R     # Reverse to get correct order.
                                      J    # Join, no spaces.
                                       »   # Join newlines.
                                        ∞  # Final horizontal mirror.