An old geezer is an old man that is 60 to 75 years old.
Sometimes an old geezer may be a cranky old geezer or just may be an old man.
The word geezer in slang means and refers to a cranky old man and can carry the connotation of either age or eccentricity.
For most people the word geezer is not offensive.
My own Grandpa called himself an old Geezer so he would not find it offensive if you called him an old Geezer.
Some old men or older people may be offended by the word Geezer but not everyone is.
Any word or saying could be offensive to some people who take offense to it while other people are not offended by those words.
The word geezer is just a word that means an old man that is age 60 to 75 years old and sometimes above.
My own neighbor who is a man is age 67 years old and he refers to himself as an old geezer and we laugh about it.
He's a nice old geezer and he's the one that brought it up first.
I'm around 30 years or so away from becoming an old geezer but I would not take offense to being called an old geezer.