Why do LCR meter manufacturers give the details of measurement frequency to us?
They give you the details, because for real-world components there isn't one concrete value; they are all dependent on the operating frequency to a greater or lesser extent.
The best frequency will depend on your application. Ideally it would be the same as your target operating frequency.
For example, if you are measuring an SMPS inductor then a high frequency will be more useful (as your circuit operating frequency is likely to be high), an electrolytic capacitor would be best measured at a lower frequency (as they don't have, and you wouldn't design expecting them to have, good high-frequency performance) and a line-frequency choke would be best measured even lower.
If we consider a real inductor there are several factors to take into consideration.
The windings have resistance
There is capacitance between turns
The permeability of the core is frequency dependent.
As such, trying to produce a single figure for inductance is difficult and the result will vary depending on the measurement frequency.
Similar arguments exist for why a capacitor or resistor is frequency dependent too.
Since the answer varies with frequency the manufacturer has to tell you the test frequency.
It is a matter of the expected user knowledge and the intended market.
Any good DMM will specify very clearly what is the operating frequency range, the measured signal characteristics, and the DMM parasitics. Some DMMs even have 'fast displays' that operate at a different bandwidth than the slower digits, these will be clearly specified in the marketing material itself. Some markings will only be understood by the experienced user, e.g.: "True RMS".
So it is not true that DMMs don't specify these characteristics, look at the low-end general purpose Fluke 114 marketing pamphlet. It is clearly seen that it is designed for signals below 1kHz. Although they do not specify loading impedance, you can go to the user's manual to find that. (BTW: the models just above this one, 115 & 117, measure capacitance by applying a known charge and measuring voltage, these do not use a frequency sweep).
But this is a general-purpose low-cost (for a Fluke) automotive and household multimeter, it is not really intended for the electronics professional. If you look further up their extensive and expensive DMM product line you will find more detailed and specialized specifications, geared towards one market or another.
When it comes to an inductor or capacitor these are far from ideal components, their actual value will depend on many factors. Temperature, DC bias, applied signal magnitude, and yes, test frequency. Depending on the application some of these factors will be important while others will not. Test instruments are designed with these factors in mind.
That is, the instrument specifications will strongly depend on the intended market. The market for most LCR meters is not your common household as it is the case for the cheaper DMMs. You cannot really compare a $10 disposable multimeter with a $6000 professional LCR meter.