Is it just a coincidence, or was there a reason why they ended in '-er'? I know that all of them derive from PIE, where they also ended in '-er'.
Also, is this '-er' the same '-er' particle, as in 'fighter', 'liar', 'runner', etc.?
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I already answerred it, so this is a copy of my previous answer:
There was the agent suffix -ter- of PIE. It was used for creation of terms for relatives and for creation of agent nouns.
Suffix -ter- was used to create a noun for person whose function or profession was to perform the action (irrespective whether he actually did it) while the o-grade of it -tor- was used to denote a person who just did the action. The combination of suffixes -a̯-ter- was used for some relatives.
Thus the term for daughter in PIE was dhuga̯tēr.
The same suffix in zero-grade, "-tr-" in combination with inanimate ending "-om" was used for creating words for tools, such as a̯ero̯trom "plow", u̯estrom "wear", tere̯trom "auger", costrom "knife".
By the way, of course, the PIE superlative -ter-os was also another use of the same suffix. Thus "gela̯teros" would mean "more joyful". The word "enter" (from Latin intra-) is another example of use of this suffix with adverbs: e̯en "in" + "-ter-om" -> e̯enterom "intestines", of which locative case is e̯enteri "inside".
The English suffex -er does not have anything in connection, it is believed to be borrowed from Latin -arius.
The Indo-European words for “father, mother, sister, brother” are all stems in –r, which appears as -ṛ, -er, -ēr, -ōr in different forms of these words.
The English agent suffix –er, as in "fighter", goes back to a proto-Germanic *-ārijaz, which is widely held to be ancient borrowing from the Latin suffix –ārius.
There is no connection between these.
This -er is no way a particle, it is probably a very ancient Indo-European suffix of unknown meaning, not connected with the modern English agent -er suffix. These words belong to the so called athematic r-stem nouns (nouns with stems ending not in a vowel, but in -r-). Not only kinship terms belonged to that noun class, the word "water" also belongs to it (Hittite 'watar', 15th century BC).