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'.

PHP, 138 133 129 bytes

PHP beats Mathematica - a rare occurence.

for(;$s=fgets(STDIN);$s>G?print$$a[$b]?$s<L:$s>L&&@array_intersect_key($$a,$$b):$$a[$b]=$$b[$a]=1)[,$a,$b]=explode(" ",trim($s));

prints 1 for truthy, empty string for falsy. Run with -nr or test it online.
needs PHP 7.1 for the list assignment; user names are case sensitive and should exclude a, b, s.

breakdown

for(;$s=fgets(STDIN);                       # loop through input
    $s>G                                        # 2. evaluate command
        ?print$$a[$b]
            # command KNOW: true if $$a[$b]
            ?$s<L
            # command SUGGEST: true if !$$a[$b] and array_intersect_key returns truthy
            :$s>L&&@array_intersect_key($$a,$$b)
        # command FRIEND: set keys in $$a and $$b
        :$$a[$b]=$$b[$a]=1
)
    [,$a,$b]=explode(" ",trim($s));             # 1. parse user names to $a and $b

• $s has to be trimmed because it includes the newline character.
• array_intersect_key has to be muted or it would yield warnings for empty $$a or $$b.
• +18 +15 bytes for all user names: Replace $$a with $f[$a] and $$b with $f[$b].

CMD (Batch), 50 + 20 + 135 = 205 bytes

• FRIEND.CMD

  @for %%f in (%1.%1 %1.%2 %2.%2 %2.%1)do @set %%f=1
  

• KNOW.CMD

  @call echo(%%%1.%2%%
  

  Prints 1 for friends, a blank line for strangers.

• SUGGEST.CMD

  @call set k=0%%%1.%2%%
  @set k=1&if %k%==0 for /f "tokens=2 delims=.=" %%f in ('set %1.')do @call set k=%%k%%%%%%f.%2%%
  @echo(%k:~1,1%
  

  Prints 1 or a blank line. I think six consecutive %s might be a new personal best.