Tuning ID-3LA RFID antenna using oscilloscope
Your suspicion is true! That 125Khz signal one may observe is the frequency from the chip. For obtaining maximum range between the RFID reader/interrogator and transponder chip, there has to be impedance match and yes, frequency match too.

One can simply use the LC tank circuit formula and and try to match it with the required frequency(If there are no internal protective circuitry and capacitors in the chip, this frequency should remain same as 125KHz.) Also in some cases the chip manufacturer would recommend ideal inductor and ideal capacitor values.

f = 1 / 2pi*sqrt(LC)

With this formula obtain the right L value for 125KHz or whichever. One can measure inductance with a simple LCR meter (available option in many advanced multimeters)

More than a decade ago, I built LF&amp;HF RFID readers with this Swiss company parts EM microelectronics. For the inductor situation (125Khz-137Khz) it was EM4095 chip. We implemented a flat spiral coil on PCB, using calculations and verified it with a simple inductance meter.

If one is also designing tags/transponders, match up the LC values to 125Khz or whichever is required.

Inductance meter is accurate and easy compared to setting up a new circuit for measuring frequency in oscilloscope and and re-calculating it for inductance. 

Also if C value is not given, one may want to check the internal parallel capacitance of that chip's coil pins too. Measure this C with an LCR meter, before soldering/wiring.