Why is that considered an inconsistency? That’s pretty much how a lot of symbols work, having multiple meanings depending on who is viewing them. For example, the Christian cross has deep meaning to a lot of people of that faith, symbolic of Christ and sacrifice, representative of an actual object in their mythology, and to a lot of people it simply IS the symbol of Christianity, one that they put on everything from jewelry to bumper stickers to coffee mugs and so on.
The symbol being the Mark of Sauron, marking his kills and his instruments (the anvil, the sword), while also having a special secret meaning to his followers isn’t weird at all.