How can chemists distinguish pure chemical element specimens that look almost "the same" as well as what deposit is what in a multimineral mined rock?
What you're describing is essentially the field of "qualitative inorganic analysis". That's a very broad subject, but a short description is that prior to the advent of advanced instrumentation, chemists developed extensive series of tests specifically to determine the elemental composition of samples.

For example, one would start with basic physical properties like color, density, and hardness, then look at reactions with heat, acids, bases, oxidants, etc. If the compound could be solubilized in any way, then one can look at how the solubility/color/etc is affected by addition of known cations and anions. If it was a geological specimen, then the location and the types of surrounding minerals would help inform the analysis as well.

A well trained inorganic chemist would have standard flow-charts of tests and could fairly confidently identify the elemental composition of common minerals and metals.