Is human mtDNA unique in its single origin?
You can trace back any piece of DNA (from any life form) to a single ancestor called LUCA (Last Universal Common Ancestor), so the short answer is 'yes'. But be careful in your interpretation.

If you look backward in time, you can see that lineages merge (siblings merge first, then cousins merge, etc...) and you necessarily end up with a single individual. In absence of recombination (which is the case for mtDNA), you can always trace back a piece of DNA this way and always find a single ancestor. The age of this common ancestor depends mainly on the population size and demographic history of the lineage. Typically, in a panmictic diploid population of constant size $N$, the expected time to the MRCA (Most Recent Common Ancestor) of the population is $2N$ generations.

So yes, you can find an "original Eve mtDNA" in any species. But please by "original" don't think this ancestor must have represented any novelty or anything of interest and do not make the mistake to think that before this ancestor no one had mtDNA.

The second paragraph is an extremely vulgarized explanation of coalescent theory. You should have a look at an intro to coalescent theory (wikipedia article here) but you might well need an intro to evolutionary biology first. Understanding Evolution might be a good way to go.