Determine the type of Cuboid

05AB1E, 26 21 bytes

None: 0 Cube: 1 Regular Domino Cuboid: 2 Shapeshifter Cuboid: 3 Brick Cuboid: 4 Floppy Cuboid: 5

•S3X@I¨%÷'•5L¦3ãPÙIkè

Try it online! or as a Test suite

Explanation

•S3X@I¨%÷'•            # push base-214 compression of the number 123224454212324512210
           5L¦         # push the list [2,3,4,5]
              3ã       # cartesian product with repetion of size 3
                P      # product of each sublist
                 Ù     # remove duplicates
                  Ik   # get the index of input in that list (-1 if non-existant)
                    è  # get the element at this index in the above number

The only place I see that we could save bytes here is find a better way of generating the number 123224454212324512210.

It is only 1-off from a prime, so one possible save would be to find the index of that prime and generate the index in less than 9 bytes.
I don't know how well the pi-function works for 21-digit primes but that could be a possibility.

JavaScript (ES6), 97 92 86 bytes

This function first checks the validity of the input, then picks the correct value from a lookup table.

Surprisingly, the longest part is the validity check (is n of the form x*y*z with x, y and z in [2,3,4,5]?). There must be a shorter way of doing it, but I couldn't figure it out so far.

n=>'NBBF..CRCC.BRR..SFRRRRR.B..C'[[34707324,0x80000800,4240,262208][n&3]>>n/4&1&&n%29]

Returns a character:

• N: None
• C: Cube
• R: Regular Domino Cuboid
• S: Shapeshifter Cuboid
• B: Brick Cuboid
• F: Floppy Cuboid

Test

let f =

n=>'NBBF..CRCC.BRR..SFRRRRR.B..C'[[34707324,0x80000800,4240,262208][n&3]>>n/4&1&&n%29]

// invalid
;[9,13,99].map(
  n => { console.log(n, f(n)) }
)

// valid
;[8,12,16,20,18,24,30,32,40,50,27,36,45,48,60,75,64,80,100,125].map(
  n => { console.log(n, f(n)) }
)