To expand on sbkgs4686's answer, I'd like to go a bit further into the history of the modern passive ending ~られる, and of the modern verb of potential, 得る【える】.
The history of ~られる
Let's start by looking at the modern passive ending ~られる and how this has developed through the years.
Monograde form
The modern passive ending ~られる has a regular 下一段【しもいちだん】 conjugation pattern -- 下【しも】 or "lower" since the stem ends in //-e// (a "lower" vowel value than the other vowel-stem ending //-i//), and 一段【いちだん】 or "one step; monograde" since the verb stem only has the one vowel for all endings.
Older bigrade form
Just like all 下一段【しもいちだん】 verbs, this comes from the older 下二段【しもいちだん】 or "lower bigrade" pattern -- 二段【にだん】 or "bigrade" since the verb stem ends in either of two vowels depending on the conjugation form, //-e// or //-u//. So modern ~られる used to have the 終止形【しゅうしけい】 ("terminal form", a.k.a. "dictionary or plain form") of ~らる.
The components: ら and る
As sbkgs4686 notes, the final ~る on the end is surmised to come from ある, the old copula ("to be") verb of existence.
- Note: The copular verb had the terminal or dictionary form of あり for much of recorded history. The form ある was the 連体形【れんたいけい】 or "attributive form", for when the verb was used to modify a noun or noun phrase.
The linking ~ら~ is of uncertain derivation; this is only used to attach the passive ending to verbs that themselves have the 一段【いちだん】 (or 二段【にだん】 in Classical and Old Japanese) conjugation patterns.
- Speculatively, I wonder if this linking element might not also be from this same ある, conjugated into the 未然形【みぜんけい】 or "irrealis / imperfective form" of [あ]ら~, as required for the passive ending, in order to explicitly add that linking //-a-// phoneme.
Before る, there was ゆ
Before there was ~らる, there was ~らゆ. This is that linking ~ら~ plus ~ゆ, the Old Japanese 助動詞【じょどうし】 or "auxiliary verb" (really, a suffix adding some kind of aspect to the verb).
This ~ゆ ending had the 下二段【しもにだん】 conjugation pattern, which later standardized into the 下一段【しもいちだん】 ending ~える. Note that this derives from ancient -yeru, with that ye sound shifting to just e through regular historical sound change. This is the same ~える we find in modern verbs like 見える【みえる】 (from 見【み】 + ゆ), 聞こえる【きこえる】 (from 聞か【きか】 + sound shift + ゆ), 覚える (from 思は【おもは】 + sound shift + ゆ), etc.
The meaning
This ~ゆ indicates that the verb action happens spontaneously (without anyone doing it), and this was also used to express passive and potential senses.
In a very similar fashion, the later ~る ending also indicates spontaneous action, and passive and potential senses.
The history of 得る【える】
Now let's look at the verb 得る【える】, commonly used in the modern language as a compounding verb to express potential.
Monograde form
Much like the modern passive ending ~られる, the modern verb 得る【える】 also has a regular 下一段【しもいちだん】 conjugation pattern -- 下【しも】 or "lower" since the stem ends in //-e// (a "lower" vowel value than the other vowel-stem ending //-i//), and 一段【いちだん】 or "one step; monograde" since the verb stem only has the one vowel for all endings.
Older bigrade form
Just like all 下一段【しもいちだん】 verbs, this comes from the older 下二段【しもいちだん】 or "lower bigrade" pattern -- 二段【にだん】 or "bigrade" since the verb stem ends in either of two vowels depending on the conjugation form, //-e// or //-u//. So modern 得る【える】 used to have the 終止形【しゅうしけい】 ("terminal form", a.k.a. "dictionary or plain form") of 得【う】.
The meaning
The potential sense of 得る【える】 is quite old, appearing already in the Man'yōshū poetry collection, finalized in 759 (with some poems written possibly centuries earlier).
That said, this verb also means "to get, to make something one's own". Consider the very many meanings of the English verb get, and how this is also used in certain combinations to express both potential and passive (potential: "I got it [VERBed]
" [I was able to [VERB]
it]; passive: "I got [VERBed]
" [I had [VERB]
done to me]; and even the causative: "I got someone to [VERB]
" [I caused someone else to [VERB]
]).
Conclusion
To sum up:
- Phonologically (in terms of the sounds), spontaneous / passive / potential ending ~[ら]ゆ and later ~[ら]る → ~[ら]れる do not match with potential / "to get" verb root う.
- There's that initial "Y" in ゆ that is absent in all attested forms of う / える.
- There's that initial "R" in る → れる that is absent in all attested forms of う / える.
- Semantically (in terms of meaning), spontaneous / passive / potential ending ~[ら]ゆ does not match with potential / "to get" verb root う.
- The oldest sense for ~[ら]ゆ expresses something that happens "on its own", while the oldest sense for う is "to get; to be possible".
Of the various components we have, and of the various historical forms we have, any overlap between modern ~られる and modern 得る【える】 appears to be purely accidental.