By the mid 1980s speech recognition was already a non-issue. Experimental system were able to handle vocabularies of 20,000 Words or more. For situations with confined vocabulary like shown in the clip every average computer with acceptable performance could do it.
Looking close reveals that this demonstration needs only a hand full of words to be recognized:
Nothing. This and the recognition of each speaker, due being trained to each one's way of speaking.
A parser would work much like known from classic text adventures. The
light torch type of games. They did allow rather sophisticated input as well. One could enter
Please turn the light on. Game wise this was not different from typing
light on - which I guess most of us did - as all the filling words are simply ignored. Same here.
And like with an Adventure game, someone demonstrating the 'astonishing' and 'intelligent' reaction of the game by use use of well worded sentences, making the conversation look more natural and sophisticated. Sales-Hokus-Pokus :)
The math behind speech recognition isn't hard at all, even an 8 bit computer can do it. The real task was to find the 'right' word to the solution computed, which means a large (*1) amount of RAM and Disk space needs to be at hand.
Such simple systems are in wide use since the mid 1980s - everyone of us has already hit those voice systems asking to say
no to a series of question or speak a number to select functions.
What differs such simple system from more modern is to guess meaning and use this in turn to rework what has been 'heard'. It takes a huge amount of data to move past detecting a few words within a confined setting. Really 'understand' what's requires and turn that into actions. There is a reason Google, with their large treasure of language sniplets, carries a leading role in today's kind of speech recognition.
Then again, the answers given in that clip are prerecorded phrases. Not because generation was complicated - Remember Speak&Spell? - but because it sounded awful due the high complexity how human speech combines words and carries over sounds. Speech output had a hype phase around that time. There were not only hardware synthesizers for next to all home computers - but as well software based solutions.
The use of pre-recorded phrases where standard in speech systems way into the 2000s. The building blocks used are clearly noticable by pauses between sentences and before the names. Different reaction to different speakers is as well easy, as the system was obvious (as mentioned by the host) trained specific to the host and the boy.
So yes, action shown was state of the art in 1984.
An April 1987 IEEE Spectrum article lists 17 commercial availabe/used systems, ranging from voice dialling to dictation, recognizing vocabularies between 13 and 20,000 words.
*1 - Large at the time. So make that a megabyte or two of RAM and 20 MiB or so in Disk space :))