Einstein proved that a magnetic field is the relativistic part of an electric field. This means that while an electric field acts between charges, a magnetic field acts between moving charges (as a charge moves through space more quickly and through time more slowly, its electromagnetic force becomes more magnetic and less electric). Therefore, the pole strength is the product of charge and velocity.
Few calculations actually involve the strength of a pole in ampere-meters because a single magnetic pole has never been isolated. Magnets are dipoles which require more complicated calculations than monopoles. However, the strength of a magnetic field is measured in teslas and one tesla is one newton per ampere-meter which confirms that the unit for pole strength is indeed the ampere-meter.
The idea that magnetic forces act on moving charges is clear in an electromagnet but not obvious in a permanent magnet. In fact, all substances have charges moving in them all of the time, one of the difficulties in reaching absolute zero. In most substances, all of the magnetic fields produced by this motion cancel each other out, but magnetic substances have more proper alignment. At the microscopic level, many molecules are magnets — hydrogen fluoride for example is a dipole. Polarity is most often expressed in terms of electronegativity but the strength of its poles could be expressed in ampere-meters as well.