Zsh, 81 bytes

typeset -A p
for x;a=(${a:#$x} ${x:|a})&&[ ${x:|a} ]&&((++p[$x]))
typeset p
<<<$a

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The expansion a=(${a:#$x} ${x:|a}) puts $x in $a if it isn't there, and takes it out if it is. Then we check if $x just got removed, and increment our pair count if it was.

With a looser definition of "list", we can shave this quite a bit.

Zsh, 56 bytes

typeset -A p l
for x;((p[$x]+=1^(l[$x]^=1)))
typeset p l

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Prints the leftover socks as all elements in an associative array with value 1, not 0.

Zsh, 81 74 bytes

-3 bytes by testing set difference in arithmetic mode instead, -4 bytes by using local instead of typeset

local -A p
for x;a=(${a:#$x} ${x:|a})&&((${#x:|a}&&++p[$x]))
local p
<<<$a

Try it online! Try it online!

The expansion a=(${a:#$x} ${x:|a}) puts $x in $a if it isn't there, and takes it out if it is. Then we check if $x just got removed, and increment our pair count if it was.

With a looser definition of "list", we can shave this quite a bit.

Zsh, 56 52 bytes

local -A p l
for x;((p[$x]+=1^(l[$x]^=1)))
local p l

Try it online! Try it online!

Prints the leftover socks as all elements in an associative array with value 1, not 0.