In a reaction between mg and hcl, the magnesium is designed to be the limiting reagent.
Not necessarily. You can design an experiment this way, but it is not a must.
Most likely the experiment you have on mind was designed this way for some specific reason. Hard to tell not knowing details.
Why is this? Is the solid always the limiting reagent?
No. For example, if you use solid CaO to absorb CO2
from the flowing gas, you want to have an excess CaO to make sure none of the CO2