"Get on" cannot be thought of as either transitive or intransitive. The property of "transitivity" applies to verbs and prepositions but not to combinations of words.
Analyzing combinations of verb and preposition as "multi-word/phrasal verbs" is misleading. To make this concept plausible, authors of traditional grammars had to introduce terms such as transitive and intransitive phrasal verbs, prepositional verbs, phrasal-prepositional verbs, which further divided into subtypes etc. No need to say that this made understanding the whole terminology a challenge on its own.
Also, under this analysis the concepts of "orthographic word" and "grammatical word" have to be kept strictly apart, the latter encompassing combinations of two or three orthographic words, which do not even have to be in sequence. This dramatically dilutes the definitions of word classes and made the very term "word" obscure.
This can be illustrated in these examples:
She brought up the child (in conversation) / She brought the child up
She brought up the child (in New York) / She brought the child up (in
She brought up the child (in her arms) / She brought the child up (in
her arms)(or: She brought the child up the stairs. etc)
In the first two the verb "bring" and the preposition "up" form an idiomatic lexical unit - obviously, we will interpret meaning 1 as "mention/introduce" and 2 as "raise". .
In terms of category, traditional grammars put the "up" in the first two sentences in the category of "adverb (and particle at the same time)". In the third one "up" is a preposition when followed by a noun phrase: She brought the child up the stairs. and adverb when it is not : She brought the child up.
The problem that this analysis causes can be clearly seen in dictionaries, which take pretty different approaches in the organization of entries.
To take as an example labeling verb-preposition combinations which cannot be interpreted on their own in isolation from nouns or other words that form the idiom .
Some dictionaries, for example, have an entry for the phrase "make up to someone" meaning "be sycophantic" (Oxford LD labels it as a "phrasal verb") but not for the similar "make it up to someone" meaning "to compensate". Others (Collins, Macmillan, Longman ..) give both these meanings under the same entry (of course, the latter meaning with the intervening "it" - "make it up to someone")
Merriam Webster labels both, "make up to someone" and "make it up to someone" as idioms. They label "bring up" as a verb, "come up to" as a "phrasal verb" , "put up with" is found under the entry for "put up", which is labeled "verb", etc.
On the other hand, what most dictionaries have in common is the general idea of "phrasal verb" as an idiomatic combination of a verb and preposition, but not the literal one. This excludes from the category the "bring up" in my third sentence .
There may be some variations in both labeling and the content of the definition. For example, "bring up" is "verb" in MW, and "phrasal verb" in other dictionaries I checked. Vocabulary.com starts the entry for "bring up" with the literal meaning "raise from lower to higher position" and continues with more idiomatic meanings: mention, raise, vomit. Oxford LD gives these three idiomatic meanings first adding to them the literal meaning of "making something appear on the computer screen" etc.
To cut the long story short, it is a blind alley to think of verb-preposition combinations such as "get on", "bring up", "come up with" etc as single words. "Transitivity" is a property shared by verbs and prepositions, but not their combinations. So we can speak of "get", "bring", "come", etc as transitive/intransitive, and we can think about prepositions up, down, on, in etc. in the same way.
"Up" in "She brought the child up" is intransitive - it doesn't have an object of its own. "Up" in "She brought the child up the stairs" is transitive, taking "the stairs" as its object.
The transitivity of prepositions has grammatical consequences - we can move the intransitive "up" next to the verb: She brought up the child. but not the transitive one * She brought up the stairs the child.
The fact that "bring up" is idiomatic in the other two examples have a strong grammatical effect of course. Most grammatical operations on idioms are simply precluded. Unlike literal interpretations, these are fixed, "take it or leave it" word combinations.