The trouble here would be dealing with more complex components.
There are two main types of electrical components. Active and Passive components.
Passive components tend to be simpler, your resisters, capacitors, transformers and so forth. These it should be possible for our engineer to work out given his multi-meter and some basic knowledge. Does it measure as having any resistance? (Resistor) Does it allow an AC current to pass? (Capacitor) Is it made of a lump of ferrous metal wrapped in two sets of wires? (Transformer) Though our engineer will probably still destroy quite a few components working this out.
Active components on the other hand are an entirely different kettle of fish. Most active components look like a blob of plastic with some metal sticking out and some markings to tell what they are. Think every thing from the classic transistor to the CPU inside your computer.
Almost every engineer when faced with a component he doesn't know will stick the code printed on it into Google to find the datasheet. Our poor engineer doesn't have this option. (Language being the big barrier here) Many many human components are encased in very similar packages, this makes standardisation of machines to produce things easier so I would expect the aliens have probably done some thing similar.
These are three components taken from my arduino starter kit. 71
One is a temperature sensor and the other two are transistors. Good luck telling them apart if you don't have Google to look up the codes.
With out knowing the correct way to wire up the thing the only options are experiment and destroy lots of them, or find one in an existing circuit and try and work out what its doing.
Neither of these options are easy or exactly safe. Who knows what the aliens make the components out of. I hope they phased out lead already for the sake of our engineer.
Having a few extra tools may make things a lot easier. An oscilloscope and power supply would be extremely helpful.
Edit: from question edit
They should be able to identify the other tools. It may be hard to tell a power supply from a multi-meter at first but it should be reasonably easy given the multimeter and a willingness to blow up a few components.
The power supply can be calibrated using some thing they can identify as a resistor and their own multimeter. Once he has worked out the voltages the dial gives out he should write a new marking sheet if it has a control nob, or a translation of the symbols for different values displayed otherwise. This will let them figure out the numbering system reasonably, which should also let them start to make more sense of other components. 71