This is all down to the fact that English is a language of acutely mongrel lineage. It has substantive roots in Celtic, Romance and Germanic languages (to name a few) and a grammar that lends itself well to the adoption of "loan words" (non-native words adopted into the native tongue.)
The "standard" means of pluralising a noun is to append -s, with some conventional variations (eg -f becomes -ves, -y becomes -ies) for convenience in spelling and pronunciation.
However latin-based words tend to pluralise in the latin fashion, so for example bacterium becomes bacteria, and cactus becomes cacti. Similarly greek-based words will adopt the equivalent pluralisation appropriate for the original root.
Still other words of Saxon or earlier origin have lovely, earthy plurals that defy the "conventions" due to their traditional forms being maintained. Geese, Mice and Children owe their unusual conjugations to their ancient roots, and to the fact that they are common words whose everyday repetition keeps them from slipping into bland conformity.
In my experience, words which do not pluralise are those which relate to herding, hunting and the counting of animals. These words tend to be saxon (germanic) or celtic in origin owing to the presence of farming and hunting in Britain long before the Norman invasion. This can be inferred by the fact that sheep, cattle and game do not pluralise, while whales, sparrows and elephants (seldom hunted or farmed in Britain!) definitely do.
I suspect these tend to be a contraction of the traditional counting forms for such cases ("head" of cattle, "brace" of partridge, "shoal" of fish) but this doesn't really answer the question of why such plurals take the same form as the singular. It could be that when counted in such a way, the animals being counted were considered an uncountable, continuous quantity (similar to water or money) that could only be "counted" when quantified with their associated counter, so cattle would be rendered an uncountable noun by its quantifying counter head. It's interesting however to note that bird pluralises to birds, while aircraft does not pluralise.
Sadly for the non-native speaker, this makes learning the "rules" of English an arbitrary and frustrating affair. However, spare a thought for the Japanese, who do not have plurals for any but a few unique nouns, and must instead learn a separate counting-suffix and corresponding character (kanji) for almost every class of noun imagineable. There are in fact entire volumes of the things, and it would be nigh-on impossible for any person to learn them all. Wikipedia lists a choice selection.
This construction is usually (or often, anyway) used in team situations:
There were more of us than there were of them.
"More of XX" meaning more people on the team. However, it is also used (perhaps ironically/humourously) when one team has only a single person. In that case, there are always more of them than of him/her.
Best Answer
There is both or why is there both would be preferable when "both" is followed by two singular terms.
I couldn't find much in the way of examples for "why is there both a... and a...", or similar uses with two singular nouns. Google NGrams was no help. But I found some for the non-question form (via Google Books snippet view):
However it is not an uncommon confusion: I found "There are 69 places in which there are both a Presbyterian and a Congregational church", which might be justified by the presence of "there are" with an obvious plural immediately before. (The Record of the Presbyterian Church in the United States of America. Volumes 24–25. Presbyterian Church in the U.S.A. 1873, page 2)
Note that "both" can be followed by two plurals, so you could have: "There are both frogs and newts living in my garden" but "There is both a frog and a newt living in my garden". (The case where you have one plural and one singular is left as an exercise to the reader.)
Also, more off-topic, but note that in colloquial English it is common to use "there's" even with a plural noun, despite it meaning "there is"; however it is less common to use "there is" with a plural. (Cambridge Dictionaries: British Grammar: There is, there's and there are).