Basically my question is that if you have a coil with N turns with its ends connected to a source of current (I), and you have another coil which is N number of single coils stacked on each other with the same current (I) passing through each of the coils, is the magnetic field strength inside both of the coils essentially the same? This is assuming that everything else besides this factor is the same.
The answer is yes. It is also the same as if you just have one solid ring (or tube) of copper with the combined current of all your N single coils flowing through it.
What the coil with n-turns lets you do is not require an unreasonably low voltage by increasing resistance or an unreasonably high current by "double dipping" or concentrating the the magnetic field of the same current flow.
I'd like to add on to the previous answer by mentioning that by definition, the two situations you described are exactly the same. A coil with N turns is the same thing (from a circuit standpoint) as N single loops in series. Due to Kirchoff's current laws, in a series circuit, the current is equivalent at all points in the circuit - meaning each turn will have the same current flowing through it.