No, it means the player is not yet out on the play.

It is used (and should be used) at all levels. Basketball is different and you shouldn't bring basketball (or football, or hockey, ...) concepts and mechanics to baseball (and vice versa)

A Fair Ball is nothing??

How often have you seen an umpire not give an indication of Fair on a ball down the line? Your suggestion that he give no signal because it's "nothing" is going to be extremely confusing. Everyone one is going to look at the umpire and wait for some indication.

When something out of the ordinary happens that is so close to call one way or the other, the umpire should give an emphatic signal when he renders his judgment. A fan reaches over the fence to grab a live ball and may or may not have touched it, it's best for the umpire to signal Safe to verify to everyone that you didn't see a touch. A batted ball whizzes by a runner and it may or may not have touched him, a Safe signal is warranted if the umpire feels the ball missed him. A fielder makes a diving attempt at a sinking line drive and he may or may not have caught it, a Safe signal is a must if the umpire believes the ball shorthopped into his glove.

By doing nothing, not only does it appear that the umpire didn't see it as Bob mentioned, but it could also affect play in a negative way. Doug Eddings learned that the hard way.

Times change. Perhaps you should consider Bob's advice. It's sound.

Here's this from J/R (see esp. #4):

E. SAFE/OUT 2.00
Safe characterizes a runner who has advanced or returned to a base before he or
the base (when appropriate) is tagged. An umpire voices and signals (or
recognizes without voice or signal) that a runner is safe.
The "safe" signal (both arms extended out to the sides, parallel to the ground) with
appropriate voice is sometimes given to indicate that
(1) an attempt to tag a runner between bases has failed (e.g., rundown1, voice- "no tag!").
(2) a batted, thrown, or pitched ball has struck a barrier but has not become dead due
to lodging, entering DBT, or fan interference (no voice).
(3) an appeal has been denied (voice- "safe" or "he's OK").
(4) interference or obstruction has been a possibility, but does not occur, or contact is
incidental: e.g., batted ball almost strikes a runner, fielder deflects a batted ball
into a runner, runner in a rundown goes out of his way to contact a fielder who is
not protected (voice in these cases- "that's nothing!").

The "play" is not over until time is called by an umpire, interference is committed by an offensive player, or the ball enters DBT. By giving the safe sign, you are communicating that interference hasn't been committed and to play on.