I am trying to figure out what to call these phenomena. For example, a sentence containing the words "specific specifications" or "participants participate", etc. Is there a word to describe this in English?
This is known as
Gift of giving
and there is a very lengthy study on the phenomena here.
The expressions you wrote are instances of Adj Noun and Noun Verb, respectively. Each expression contains words derived and inflected from a given root. For instance, the lexical morpheme for your first expression is specify (or possibly spec) and for the second, part. With these base words, we can construct a class of words, known as a word word family by attaching to them derivational morphemes, which modify the word's part of speech or meaning OR inflectional morphemes, which modify tense and number. So for instance the word family of spec(-ify) would be specification, specifier, specified, etc. Because a word family contains different parts of speech, a single word family can generate grammatically structured expressions like the ones you wrote. As far I know, there is no term for such expressions in English, but since roots encode semantic information, it's not surprising that the grammatical expressions derived from a word family are often tautological.
One of your examples was as follows.
Individually, these words have the following definitions.
Clearly defined or identified.
A detailed description of the design and materials used to make something.
Together, they can be interpreted as follows.
A specification that is specific to something in the full sentence or context.
Words diverge and evolve. I don't know of any special term for these "dual word same root" combinations. Neither do I see any point in coining such a term.
Sounds like tautology to me, though let's wait for 100% sure answer.