I noticed in the Google dictionary there isn’t a differentiation between pigeon (Taube) and dove (Taube). Is this true, that there is one word for two very different species?
Yes, this is true. You’ll often find words with two meanings. I guess this is true for lots of languages, not only German. I met such words in French and English for example.
Look here:
ESL Quiz - One Word -- Two Meanings (Letitia Bradley) I-TESL-J (English)
Words with More Than One Meaning | Learn English (English)
http://home.alphalink.com.au/~umbidas/homonym_main.htm (English)
Using Der, Die, and Das Correctly in German (German)
http://germanheit.com/post/2637134772/words-with-two-or-more-meanings-1 (German)
“Tauben” is the name of a family, so it includes many different species. Each of this species has their own German name. For example the most common “Tauben” you will see in towns are the “Felsentauben”/“Stadttauben”.
Likewise in English “dove” and “pigeon” are no species names. It is not even clear which species belong to the doves and which are pigeons. From Wikipedia:
"Pigeons and doves constitute the bird clade Columbidae that include some 300 species of near passerines. In general the terms “dove” and “pigeon” are used somewhat interchangeably. In ornithological practice, there is a tendency for “dove” to be used for smaller species and “pigeon” for larger ones, but this is in no way consistently applied, and historically the common names for these birds involve a great deal of variation between the terms “dove” and “pigeon.”
That’s interesting, nobody here where I live would call a pigeon a dove. I never heard of the Wikipedia definition, where I live it’s pretty obvious the different between the two as there’s not that many different species here where I live, a dove is an elegant looking, smaller white bird and a pigeon is a totally different looking bird. I guess I’ll have to look it up myself, guess I’m learning something new. Maybe its like a bear? Here in Canada we have several species, black bear, polar bear, mountain grizzly, coastal grizzly and tundra grizzly, the polar bear, black bear and grizzly are very different looking bears, but are all bears.
In high school one of my best friends (a Cuban refugee, so a native Spanish speaker) owned an old, white Ford Falcon which he called “la paloma.” As he put it one day when we were laboring up a long hill in the car, “On the good days, when this car can make it up these hills, I think of it as ‘my dove.’ On days like this, it’s ‘the pigeon.’”
In my opinion, this is more a question of definition than the fact that “a word can mean two things or more” (as suggested by Vera’s links). Swedish has the same base word for “turtle” and “tortoise” (the main difference is whether the reptile dwells on land or in water)
Animals, flowers, trees and colours seldom match 100%.
Little of anything matches 100% 
p.s. I have no idea of the differences between a dove and pigeon.
I shouldn’t post so early in the morning. Jeff is right. I overread the meaning of the question. I read “Taube” and associated it immediately with the two different meaning of this word in German.
Neither do I know the difference between a parrot, a parrakeet, a lorikeet, a rosella and a cockatoo!
True, very few things exactly match between languages.