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On a web-show called Bravest Warriors there is a character called Paralyzed Horse. He has a tendency to make these interesting monologues and I was curious if there was a name for the kind of language or technique he uses when he delivers these monologues.

Here are a couple excerpts:

Paralyzed Horse's log, July 3, 3085.

Everyday I hear and I see eternity. I am frozen in awe of my knowledge of forever, but I belong only to one girl. Nowhere can I see more clearly than into her heart. When Beth feels heartbreak, I'm stranded in the Stygian Void. Helpless to comfort her, for I am just a horse. Sleep now, Bravest Warriors, and dream of doors and keys and Ralph Waldo Pickle Chips. Today something unnatural has slipped into your universe, an octave of death, a tentacle of time. It is the final gift and it is not for you. Beth does not know it now, but this gift is for her.

https://youtu.be/Sh2SM3cnmaM?t=32

Additionally from a series called Watchmen, a character named Rorschach delivers a similar style/tone:

Rorschach’s Journal. October 12th, 1985:

Dog carcass in alley this morning, tire tread on burst stomach. This city is afraid of me. I have seen its true face. The streets are extended gutters and the gutters are full of blood and when the drains finally scab over, all the vermin will drown. The accumulated filth of all their sex and murder will foam up about their waists and all the whores and politicians will look up and shout “Save us!”… and I’ll whisper “no.”

http://www.monologuedb.com/dramatic-male-monologues/watchmen-rorschach/

If there exists a name for this I would love to know.

This style is called stream of consciousness:

a style in literature that is used to represent a character's feelings and thoughts as they experience them, using long, continuous pieces of text without obvious organization or structure

Cambridge English Dictionary

Rorschach's journal is more obviously an example of this. The "Paralysed Horse" narrative is, perhaps, too carefully constructed to really count as stream of consciousness, but it is certainly affected by, or derived from, the style.

It is a relatively modern concept, especially its use in literature:

The term was first used by the psychologist William James in The Principles of Psychology (1890). As the psychological novel developed in the 20th century, some writers attempted to capture the total flow of their characters’ consciousness, rather than limit themselves to rational thoughts.

Encyclopaedia Britannica

It has been used by many 20th century writers such as James Joyce (eg. Ulysses) and Virginia Woolf (eg. Mrs Dalloway and The Waves).