Learn English – name for when the same word is used at the beginning and end of different phrases/sentences

You are describing word association football, a variation on the classic rhetorical device anadiplosis, “the repetition of the last word (or phrase) from the previous line, clause, or sentence at the beginning of the next”.¹ In this variation, anadiplosis is combined with ellipsis, “omission of a word or short phrase easily understood in context”.²

Word association football is essentially surprising in a humorous and literary sort of way, like a pun. Your examples are typical.

While not necessarily the originators of this rhetorical device, Monty Python get widespread credit for naming it.⁴ The name stems from one of their monologues.

Here is an example of anadiplosis, followed by the canonical example of word association football. Bolding, italics, and line breaks are added to emphasize the rhetorical structure.

Glad You Came (example of anadiplosis)

Turn the lights out now
Now, I'll take you by the hand
Hand you another drink
Drink it if you can
Can you spend a little time
Time is slipping away
Away from us, so stay
Stay with me I can make
Make you glad you came
—The Wanted³

Word Association Football

Tonight’s the night
I shall be talking about of flu
the subject of word association football.
This is a technique out a living
much used in the practice makes perfect
of psychoanalysister and brother
and one that has occupied piper
the majority rule
of my attention squad by the right number one two three
four the last five years to the memory.
It is quite remarkable baker charlie
how much the miller’s son
this so-called while you were out
word association immigrants’ problems
influences the manner from heaven
in which we sleekit cowering timrous beasties all-American
speak, the famous explorer.
And the really well that is
surprising partner in crime
is that a lot and his wife
of the lions’ feeding
time we may be c d e
effectively quite unaware of the fact or fiction section of the Watford Public Library
that we are even doing it is a far, far better thing that I do now then, now then, what's going Onward Christian Barnard the famous hearty part of the lettuce now praise famous mental homes for loonies like me.
So on the button,
my contention causing all the headaches,
is that unless we take into account of Monte Cristo
in our thinking George the Fifth
this phenomenon the other hand
we shall not be able satisfactor fiction section of the Watford Public Library again
ily to understand to attention when I’m talking to you and stop laughing
about human nature, man’s psychological make-up some story the wife’ll believe
and hence the very meaning of life itselfish bastard, I’ll kick him in the balls Pond Road.
—Monty Python