The following was explained to me by the attendant the last time I gave blood;

We have been at war - (and still are?) and there are also countless natural and man made disasters the Red Cross is involved with.  As a policy the Red Cross doesn't show up on caller ID because the name could cause undo alarm to a person who has a relative, friend or other.

Sounds okay to me - I've labeled the number as the Red Cross on my phone