Nor for copyright, but having copyright does not guarantee legal protection for copyright. If you can't prove that you have it and had it first, you may as well not have it at all.
The site you linked to says quite clearly that you should register to get:
"""
# a legal presumption of the registrant's ownership of the mark and the registrant's exclusive right to use the mark nationwide on or in connection with the goods and/or services listed in the registration;
# the ability to bring an action concerning the mark in federal court;
"""
So in other words, if you don't register, you don't have these things. Yes, you have a trademark, sure... you just can't do anything about it.
"The owner of a federal trademark registration may sue in federal court more easily. It is worth noting that even without a federal trademark registration you still may be able to sue in federal court, but it may be more costly when proving jurisdiction (i.e. diversity of citizenship & amount in controversy over $75,000). Even if you can't sue in federal court, you can always sue in state court. However, attorneys usually prefer federal court in trademark cases for perceived procedural advantages."
The site you linked to says quite clearly that you should register to get:
""" # a legal presumption of the registrant's ownership of the mark and the registrant's exclusive right to use the mark nationwide on or in connection with the goods and/or services listed in the registration; # the ability to bring an action concerning the mark in federal court; """
So in other words, if you don't register, you don't have these things. Yes, you have a trademark, sure... you just can't do anything about it.