Begegnen usually has a subject and a dative object. The subject specifies the occurance, the dative object the experiencer.
Wer hier in die Tiefe taucht, dem begegnen leuchtende Clownfischchen, ... (as an alternative to the DWDS example, above @DavidVogt)
However, there is another variant that has a rather abstract idea that appears as the grammatical subject. It works basicly the same:
Dieses Motiv begegnet ihm auch in der altorientalischen Bildkunst.
Except that the dative object can be implied:
Dieses Motiv begegnet auch in der altorientalischen Bildkunst. (bibelwissenschaft.de)
This is the way the verb begegnen is used in your example.
Außer Formen vorübergehender Einsamkeit […] begegnen auch solche [Formen der Einsamkeit], die …
However, the problem is figuring out the experiencer. Ellipsis isn't always the correct analysis.
Dieses Motiv begegnet ihm auch in der […] Bildkunst.
It would be grammatical for example to infer that Picasso encountered the violin in his blue phase as a point of attraction, or: Die Violine begegnete Picasso auch in seiner blauen Phase. This is stressing that I have once learned something about cubism more than it says about violins or the blues. There is a fine difference.
abstract: denn er sprach, es möchte im ein unfal begegnen. 1 Mos. 42, 4; (DWB)
Compare widerfahren, ihm wird ein Unfall widerfahren. Or become, he will become ill, that gown becomes you.
Many more such examples in Grimm that would not occure to me.
hervorzuheben ist die bedeutung von geschehen, sich zutragen, ereignen, engl. occur, meist ohne dativ:
ich bitt um gnad, es soll nicht mehr begegnen. Schiller 536 (DWB)
[DWB] https://www.dwds.de/wb/dwb/begegnen
From comments
I guess when you translate this entire sentence into English, you can't translate the word "begegnen" as "encounter" in any sensible manner, can you?
You can verb any driveway you want but I hope to have shown that it is not always a good choice.
It isn't very difficult as long as fish aren't anthropomorphic. Thus the cited example:
Wer hier in die Tiefe taucht, begegnet leuchtenden Clownfischen, exotischen Zackenbarschen und märchenhaften Korallen. (DWDS begegnen)
But I do wonder if an editor had to cut words at that point (sometimes to fit a collumn):
Außer den Formen vorübergehender Einsamkeit, die sich aus schnell wechselnden Bedingungen inneren oder äußeren Lebens ergeben, begegnen [wir] auch solche[n], die schon größere Zeitabschnitte prägen und solche[n], die weitgehend fixierte Endzustände des Einsamkeitsbewußtseins sind.
Although it is true that the verb would be singular in the most likely event as David Vogt has shown, as is further suggested by agreement with the relative clause, die sich ... ergeben, the assumption that the numerus is inflected for the enumerated forms is nevertheless not parsimonious because it could be put in singular. The difference of solche[n] is minimal to me and the origin of Formen is forma, ironically.
Grimm even has examples of what looks like nominative Sie (cf. 3). So, ? Picasso begegnet die Violinen.