This is not unusual, and having the fence entirely on your side means it is your fence and there is no question of disputes with current or future neighbors about who owns it. My rear neighbors have done the same thing.

I do not think a neighbor can claim possession of that strip simply because you have placed your fence well inside the property line.

As to who put the markers in, if they are permanent metal rods or posts, they were installed by the surveyor who laid out the lots. If they were just temporary wood stakes, they could still be a surveyor but also could be builder's markers.

I assume the fence builders told the neighbors also, but you might just have a friendly visit with them and let them know what the fence guys told you, so they know that you know where the line is.

Good fences and friendly communication make for good neighbors.

Thanks for the reply, I feel better now. I really did not want to replace the back fence.
Have a great day!

Make sure you are the one who mows and weedeats that strip of land. assuming they didn't put up an entire fence and just butted up to yours that means they would technically have installed a portion of their fence on this property with your knowledge, and so if they also may be able to claim at least an easment to that strip of property after a number of years, especially if they are the ones maintaining and using it.

I would check out the property laws in your state, and if necessary ask your neighbors to move any part of their fence off your property.