#Pyth, 71 bytes
x[.EmqQ*shdsed.cs.pM.cK`Q/lK2 2qsP{*MyPQQ&P_hyQP_Q<.&QhQP_QqQsm^sdlKK)1
Try it here! or Verify the test cases.
Alternative solution, 71 bytes:
x[.EmqQ*shdsed.cs.pM.cK`Q/lK2 2qsP{*MyPQQ&P_hyQP_Q&.AjQ2P_QqQsm^sdlKK)1
This returns -1 for False, 0 for Vampire, 1 for Perfect, 2 for Sophie Germain, 3 for Mersenne and 4 for Narcissistic. In case more are truthy, this picks the one with the lowest index. Unfortunately, this memory errors for the following test cases: 33550336, 16758243290880.
#Explanation
##Vampire (≈ 30 bytes)
.EmqQ*shdsed.c.nsM.cK`Q/lK2 2
.c `Q/lK2 Get all combinations of the input's digits of half its length.
.pM Get all possible permutations of each.
.cs 2 Get all possible two-number combinations, flattened.
.EmqQ*shdsed Check if any has the product equal to the input.
##Perfect (10 bytes)
qsP{*MyPQQ
q Q Is equal to the input?
s The sum of:
{*MyPQ Its divisors,
P Popped (since we want the proper divisors).
##Sophie Germain (9 bytes)
&P_hyQP_Q
P_Q Is the input prime?
P_hy Is the input doubled + 1 prime?
& Logical AND.
##Mersenne (9 bytes)
<.&QhQP_Q
< Is smaller?
.& Logical AND between:
Q The input
hQ And the input + 1
than:
P_Q Is the input prime?
In the alternative approach, this part is changed to .AjQ2P_Q.
##Narcissistic (10 bytes)
qQsm^sdlKK
q Is equal?
Q The Input
m^sdlKK Each of its digits raised to the power of the length
s Summed.