What is a proper word for "manner of speech"?

There are many words, here are a couple of them:

phraseology
1.the manner in which words or phrases are used
Collins Dictionary

1.manner or style of verbal expression; characteristic language:
legal phraseology.
Random House Kernerman Webster's College Dictionary

A particular mode of expression, especially one characteristic of a particular speaker or subject area.
‘I know that my phraseology and way of putting things is sometimes a bit direct.’
Oxford Living Dictionaries

There is also "register". "register" doesn't necessarily have to do with formality, class and social status, but it very often does. It's also a bit more jargonistic (used more often in the field of language study)

1.(Linguistics) a form of a language associated with a particular social situation or subject matter, such as obscene slang, legal language, or journalese.
Collins Dictionary

9.A variety or level of language used in a specific social setting: speaking in an informal register; writing in a scientific register.
American Heritage Dictionary

Finally, there is also "diction", and there are variations in exact meaning among dictionary definitions for this. For example some stress clarity and distinctiveness of pronunciation, and some just describe it as manner:

diction
1.(Linguistics) the choice and use of words in writing or speech
2.(Phonetics & Phonology) the manner of uttering or enunciating words and sounds; elocution
Collins Dictionary

1.The choice and use of words and phrases in speech or writing.
2.The style of enunciation in speaking or singing.
Oxford Living Dictionaries

n.
1. Choice and use of words in speech or writing.
2. Degree of clarity and distinctness of pronunciation in speech or singing; enunciation.
American Heritage Dictionary

The closest thing to gait for speaking that I can think of would be the noun articulation. That might be the word you're looking for. Simply put, someone who has very poor articulation is not able to speak very well or express themselves in a clear and coherent way using their facility of speech. Most dictionaries would define this word similar to something like this (this particular definition has been taken from the Cambridge Dictionary):

the way in which you pronounce words or produce sounds

Example sentence:

A good singer needs to have good articulation (= a clear way of pronouncing words).

Thus, you could refer to somebody's manner of speech as their manner of articulation or the manner in which they articulate their speech.

Just as an additional remark, Wikipedia even has an entire page dedicated to articulation where you can go and learn everything you want to know about this term:

Articulation is the movement of the tongue, lips, jaw, and other speech organs (the articulators) in ways that make speech sounds.