What is technical difference (internal voltage) between Digital multimeter's current measuring ranges mA (milliampere) and A(ampere)?
Nearly all DMMs are in essence volt meters.

What can you do if you only have a volt meter but want to measure a current? You use a resistor. In measurement terms we call that a shunt resistor and the current you want to measure needs to flow through it.

What if we want to measure 1 A?

We could use a 0.1 ohm resistor and then we would measure 0.1 V across that resistor.

What if we want to measure 10 mA?

We could use a 10 ohm resistor and then we would measure 0.1 V across that resistor.

So your DMM very likely only uses a different value shunt resistor for the mA and A range. Optionally there might be a (different) fuse or no fuse (for 10 A or more range for example) for each range.

The brownout you get because of the supply voltage becoming too low. If you measure in the mA range the shunt resistor has a higher value compared to measuring in the A range. At startup your product might for a short time use a lot of current. That high current across the shunt resistor for the mA range can result in so much voltage drop (burden voltage) that the supply voltage for your microcontroller becomes too low and the brownout detection triggers.

Bonus info: Here's an article describing a typical DMM from the end of the previous century, using an ICL7107 IC. Note how for the current ranges each range uses a different combination of resistors.