Answer: 40

Small Hint:

Sort the three side lengths of the smaller prism and compare them in order with $a,$ $b,$ and $c$

Big Hint:

The two unchanged dimensions force $\frac{a}{1995}=\frac{1995}{c}$

Solution:

Let the similar smaller prism have sorted sides $x\leq y\leq z.$ It shares two side lengths with $P,$ and all three of its sorted sides are smaller than the corresponding sides of $P.$ The only possible matching is $y=a$ and $z=b=1995.$ Similarity then gives $$\frac{x}{a}=\frac{a}{1995}=\frac{1995}{c},$$ so $ac=1995^2.$ Conversely every factor pair $a<c$ gives a nondegenerate cut. Since $1995=3\cdot5\cdot7\cdot19,$ its square has $3^4=81$ divisors. Excluding the central pair $a=c=1995$ and taking one divisor from each remaining pair gives $\frac{81-1}{2}=40.$