Should we be friends?
SWI-Prolog, 62 47 41 bytes

X*Y:-X+Y;Y+X;X==Y.
X?Y:-not(X*Y),X*Z,Y*Z.


Prolog isn't too often useful, but when it is it's just beautiful. We'll use a+b to notate that a is friends with b, a*b that a knows b and a?b that b should be suggested to a or not. The first line simply says that X*Y is true if either X+Y, Y+X or X == Y is true. This implements the symmetry of knowing eachother. Asking if there should be a suggestion is incredibly simple. We just ask if there is a Z such that X*Y is false and X*Z and Y*Z is true. Exactly as described in the challenge.

If you save this as a file (e.g. friends.pl) and open SWI-Prolog with this file (prolog -l friends.pl) you get dropped into a REPL.

You can assert friendships like this:

assert('a' + 'b').
assert('a' + 'c').
assert('b' + 'd').


You can check if people know eachother or suggestion should be made:

'a'*'b'.
'a'?'d'.