I want to describe words which produce pleasure effect as they are similar sounding.
For example, I want to describe the similarity in the pronunciation of
- Vinni, Vijji, Vikki
- Amit, Sumit
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First, we have to leave out the "pleasing way" part of the question; pleasure is an esthetic judgement and that's not language, that's taste. Let's talk about what sounds can go together in words; that's language. In this case, English language.
Second, it matters what language the listener is hearing in. Clusters that are normal in English can sound very surprising to listeners of other languages.
That said, both assonance and rhyme (spelled 'rime' in the technical literature) are at least some of the words you're looking for.
English words are made up of syllables, just like words of any language; but syllables vary wildly from language to language. There are so few possible Japanese syllables that kana of about 50 symbols can represent them all, but if there were a kana for English, there would have to be over 25,000.
Take the word stump, for instance. It starts with a cluster /st-/ -- that's the assonance -- and then a vowel followed by another consonant cluster /-əmp/ -- that's the rime. It's very simple:
Assonance plus rime equals syllable.
There are lots of complicated rules that describe how sounds can go together in big words, but 1-syllable words are where the interesting things happen. It turns out that words with particular assonances and rimes tend to cluster semantically, too.
ST- assonances tend to refer to long thin (one-dimensional) rigid objects --
stick, stiff, stem, stand, stab, stud, etc -- while words with -əmp rimes tend to refer to lumpy, bumpy, humpy things that are three-dimensional, with all dimensions roughly the same, like dump, rump, slump, clump, etc.
These senses are both there in the word stump -- it's an -əmp word, it's the right shape; but it used to be a tree -- which fits the st- sense, too.
There's a lot more research available on this topic, by the way.
The right term in this context is assonance, which according to Merriam-Webster means "relatively close juxtaposition of similar sounds especially of vowels; repetition of vowels without repetition of consonants (as in stony and holy)." Compare with MW's definition of consonance "correspondence or recurrence of sounds especially in words; specifically : recurrence or repetition of consonants especially at the end of stressed syllables without the similar correspondence of vowels (as in the final sounds of “stroke” and “luck”)."
Rhyme could refer to similarity in last syllables of two words. Eg: "Amit" and "Sumit" rhyme with each other, as do cat and hat.
Alliteration could refer to a rhyme in the first syllables of two words. Eg: Dirty Dancing and King Kong are alliterations.