Very straightforwardly: the two terms come from mathematics.
"Subscripts are often used to refer to members of a mathematical sequence or set or elements of a vector. For example, in the sequence O = (45, −2, 800), O₃ refers to the third member of sequence O, which is 800." (Wikipedia)
As fanf2 said, projection in software typically refers to selecting one element of a tuple.
"[Projection is] An operation typified by the jth projection map, written projⱼ, that takes an element x = (x₁, ..., x ⱼ , ..., xₙ) of the Cartesian product X₁ × ⋯ × Xⱼ × ⋯ × Xₙ to the value projⱼ(x) = xⱼ" https://en.wikipedia.org/wiki/Projection_(mathematics)
"Subscripts are often used to refer to members of a mathematical sequence or set or elements of a vector. For example, in the sequence O = (45, −2, 800), O₃ refers to the third member of sequence O, which is 800." (Wikipedia)
As fanf2 said, projection in software typically refers to selecting one element of a tuple.
"[Projection is] An operation typified by the jth projection map, written projⱼ, that takes an element x = (x₁, ..., x ⱼ , ..., xₙ) of the Cartesian product X₁ × ⋯ × Xⱼ × ⋯ × Xₙ to the value projⱼ(x) = xⱼ" https://en.wikipedia.org/wiki/Projection_(mathematics)