I was trying to determine how many people with the same name and the roughly same age were living in that part of New York. That could help decide the odds for the 1864 enlistment being your Frederick.
Could this be your Frederick in 1860, rather than the one you have noted:
1860 New York Ward 8 District 1, New York, New York
Frederick Meyer, age 40, born about 1820 in Germany, Grocer
Dora, wife, age 36, born about 1824 in Germany
Theodore Goldanstine, age 27, born about 1833 in Germany, Clerk
In New York City in 1860, there were 6 Frederick Meyer's born within 5 years of 1825.
For Brooklyn, 1860, there was one Frederick Meyer, born 1820.
There weren't any born around that time with Meier spelling.