Testing regulated power supply: 100 mA current sink
At 5V you should probably be OK. But who knows what the safe operating area is for those transistors... Why not use a 2n3055 after the 2n2222? Then the 2n3055 will never fail at least up to 10V and 2A. Also, be very careful if you replace the transistor with a mosfet in any electronic load. Check the DC SOA curve of the mosfet. [At 10V and 200mA, probably even a BD139 will be fine, and has a hole to attach a heatsink.] Make sure it is in a darlington configuration with the 2n2222, so base current (op amp output) is minimal and will not throw off readings.

As for your question, LM358 has a fairly high offset voltage. Additionally, load resistors will not be matched. So the op amps will settle at different voltages across the resistors. Meaning, different currents through the transistors. Should probably not cause a thermal runaway (as long as transistor SOA is maintained at DC). But, probably better to use a single op amp if you want any reliability between readings and linearity (i.e. single op amp driving 2 transistors, each with its own separate load resistor at emitter).

Why is it a problem? Probably not a problem if an ammeter is permanently attached to collector. Although, it will never be as accurate as placing ammeter at emitter, due to base current coming from op amp. Also, the components will have different temperature characteristics, so results may not be repeatable with respect to the position of the reference pot wiper position. Also, it is mighty inconvenient if you just want to measure voltage at the load resistor and calculate current, since there are two of them.

Since the circuit aims to calculate microcontroller current, I'm guessing accuracy is a requirement, if not at uA range, then at least in 0-5mA range. Probably best to use a TIP122, or a really suitable mosfet rated for linear use. If the circuit is never ever operated at voltages above 10V, an IRFZ44N might be fine up to 1A See my question here. Since no gate current flows into the source, this will give most accurate result.

PS: also, your schematic shows the LM358 running from the same supply. I'm assuming you have a separate regulated source for the LM358 and reference pot.... i.e. they are not connected to the transistor collectors which should be at the power supply under test.