The question is a bit vague (not that that is all bad) so my answer will be as well. There are (infinitely) many ways to define sequences and matrices of numbers and polynomials. Some of the simpler ones lead to sequences with special names. Simple transformations will send some of these nice ones to others. A sequence of polynomials gives an array of coefficients . Diagonals can give something else. For example the binomial coefficients have their recurrence $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$. If we change it to $B(n,k)=s\,B(n-1,k-1)+B(n-1,k)$ (with appropriate initial conditions) then we just have $B(n,k)=\binom{n}{k}s^k$ and each row can be viewed as a polynomial $(1+s)^n$. It is well known that the Fibonacci numbers can be viewed as sums of certain diagonals of Pascals triangle. If we use instead the B(n,k) then we get the Fibonacci polynomials which start your question. $1,1,s+1,2s+1,s^2+3s+1,3s^2+4s+1\dots$ Splitting these into even and odd terms gives us $1,s+1,s^2+3s+1,\dots$ and also $1,2s+1,3s^2+4s+1,\dots$ either is a basis of the space of polynomials as is $1,s,s^2,\dots$ sending any one of these to an integer sequence (starting with 1) sends the others to another. See what the linear transformations in your examples do to $1,s,s^2,s^3,\dots$
Here is another example (which I was happy to discover) My personal favorite recurrence is that for convergents to $\sqrt2$, $1/1,3/2,7/5,17/12,41/29,\dots$ The numerators $1,3,7,17,41,\dots$ and the denominators 1,2,5,12,29,... both satisfy the recurrence $a_n=2a_{n-1}+a_{n-2}$ just with different initial conditions. They could be called the Pell and Pell-Lucas numbers. If we say $A_n=2A_{n-1}+sA_{n-2}$ then we get a sequence $1,1,s+2,3s+4,s^2+8s+8,5s^2+20s+16,\dots$ These are closely related to Chebyshev polynomials of the first kind.(replace s by -s^2 and multiply every other one by s). Avoiding many digressions, Consider the two sequences
$$1,s+2,s^2+8s+8,s^3+18s^2+48s+32,\dots$$ $$1,3s+4,5s^2+20s+16,7s^3+56s^2+112s+64,\dots$$
If the first is mapped to 1,0,0,0,0...then the last is mapped to 1, 2/3, -8/15, 32/21, -128/15, 2560/33, -1415168/1365, 57344/3, -118521856/255 which is intriguingly close to the negatives of the cosecant numbers 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255
If the last one is mapped to 1,0,0,0,0 then the first is mapped to -2,16,-272,7936,... which are the tangent numbers (but with alternating signs).