I think the term you're looking for describing the nature of the interests is "obsessive". Quoting from a presentation aimed the general public:
Why so obsessive? Exactly why special interests, obsessions and autism tend to go together is still not very well understood. However, if you look at the key characteristics of autism, it does make a certain sense.
To begin with, autism is associated with ‘rigid’ thinking. This sounds quite judgemental, but another way to say it would be to say that people with autism tend to be more all-or-nothing than most people: something is either absolutely not interesting or absolutely fascinating.
This can be an oversimplification, but people with autism do tend to think in fewer shades of grey. When applied to favourite subjects, this can turn an interest into a passion, or an obsession.
And from a more DSM-ish perspective (that is not presupposing a common cause) that is stated as
Research suggests that anxiety disorders and OCD are highly prevalent in individuals with ASD. However, the significant overlap of ASD features with anxiety and OCD symptomology makes differential diagnosis of these disorders particularly challenging.
In the autism literature, the term "obsession" is used
to indicate strong, repetitive interests, even if this is not accompanied by anxiety.
The latter paper also has a survey of the content of obsessions in children with autism spectrum conditions, in case you're interested. (Generally speaking, such obsession tend to be more object- rather than people-oriented in subjects with autism.)
Keep in mind that the distinction between a "passion" and an "obsession" can be a bit value-laden, and not just when talking about autism, but also in the wider context of perfectionism etc.
An alternative term with practically the same meaning as "obsessive" [interests] is "circumscribed interests". I don't read enough autism literature to tell you if this or "obsessive" is the more prevalent term used. It's probably wise to search for both. It doesn't seem like the terminology has settled. The DSM is not terribly helpful in this respect because it discusses/titles the notion together with behaviors, more precisely using the term(s) Restricted and Repetitive Behaviours and Interests (RRBIs).
Finally the "rambling" part of your question relates more to social deficits present in autism, so what you're asking for is a combination of terms/notions rather than just one. In fact these two dimensions (social deficits and RRBIs) are considered definitory for autism... so the umbrella term you're looking for (for someone rambling about their very narrow interests) is basically... "autistic".
It would be nice if there were a term for some subset of A (social deficits) + some subset of B (RRBis) features, but I doubt you'll find a term so specific given the state of the literature I've looked at. So if "autistic" comes off as too broad, the precise answer is that there isn't a specific term for what you're asking... unless you want to coin/combine existing terms.
It's rather hard to even find a technical term just for the autistic rambling (regardless of subject). I did find an informal term for this used by autistic persons themselves, which has been reproduced in a psychiatry monograph, namely "data dump":
Paradoxically, when patients do eventually begin to talk, they can soon switch into a speech-like monologue that is difficult to interrupt or halt, and may seem rambling. (Often they are also quite self-conscious about how much they are digressing and 'wasting [your] time'.)
Being autistic means being slow to notice non-verbal cues from another person [i.e. from you the doctor] that signal to them [the patient], 'it's my turn to say or ask something now'. It is as as if they have been silently building up their in their minds a long and very detailed account of their lives, which is suddenly released. Some on the autism spectrum refer to this using the computer analogy of 'data dump' (there is no humorous intent here). Attempts to interrupt this flow can produce irritated reactions. This is not anger (although expression of anger can occur suddenly). A common complaint of people on the autism spectrum is that interruptions are hugely problematic--in more extreme cases the only way to resume is to start over at the beginning. Equally, they are sometimes aware of this. 'I know I talk on too much, but I just don't know when to stop, the way other people seem to know with its time to give way'. [sic]
So... if we were to coin a combined term for precisely what you are asking for... "obsessive interest data dump" (more informally) or "circumscribed-interest speech-like monologue" (somewhat more formally) would be the term combo I'd use. And that's for the action itself; for the [pre]disposition to it; prepend that word.
Finally the notion of perseveration defined as
the continuation or recurrence of an experience or activity without the appropriate stimulus
does seem to apply to the [socially uncalled for] "data dump" as well, but perseveration is a slightly broader term, so "perseverative speech/discourse on fixated interest" seems more precise... although there are at least some DSM-5 guidelines on autism using the terms "perseveration on a specific topic" (under criterion B3) and "preservative speech" (criterion B1), and it seems they suggest the former (rather than the latter) for what you ask about. (criterion B3). Note however that "failure of normal back
and forth conversation through reduced sharing of interests" including "One‐sided conversations/monologues/tangential speech" is under criterion A1... and (all three) A's and two B's are required for a diagnosis under DSM-5; and they also suggest to "avoid using the exact same behavioral exemplar to satisfy two criteria." (In general, meeting the DSM-5 criteria for autism is more difficult than for DSM-IV-TR; furthermore, the tightened criteria also results in greater OCD comorbitidy.)