This answer is going to be less useful than I would wish, because I don't have a reference. But I can at least tell you the answer.
These knots are clearly all elements of the braid group on two strands, called $B_2$. Elements of $B_n$ are generated by taking $n$ separate strands, switching the places of one end of each a pair of strands without switching the other ends,thus half-twisting the two strands together. The group $B_2$ on two strands is particularly simple: it is isomorphic to the integers. We can identify its elements just by saying which integer it corresponds to. This is simply the number $k$ of half-twists of the two strands, which is the same as the number of half-twists of your strip. (Or in the case of negative $k$, half-twists in the other direction.)
If you take a strip and give it $k$ half-twists before gluing the edges, the result is a knot with one component if $k$ is odd, two components if $k$ is even.
When $k$ is odd, the knot is an unknot when $k=1$, a trefoil when $k=3$, a cinquefoil when $k=5$, and so forth. Knot notations usually express the common structure of this family of knots. For example, in Conway notation these are $[1], [3], [5], [7],\ldots$. They are all torus knots (meaning they can be embedded in a torus) and in the special torus knot notation they are written $(2, k)$.
When $k$ is even the edge forms two linked circles, linked $\frac k2$ times. (Or in the trivial
$k=0$ case two unlinked circles.) For $k=2$ this is the Hopf link, two circles linked in the simplest possible way, and for $k=4$ it is sometimes called Solomon's knot, just like the Hopf link except linked $\left(\frac k2=2\right)$ twice instead of once.
These are also torus knots, again written $(2,k)$ in the special torus knot notation. There is a theorem of torus knots that $(a,b)$ has a single component exactly when $a$ and $b$ are relatively prime, so for your family of knots, there is a single component exactly when $k$ is relatively prime to $2$; that is when $k$ is odd.
It might also be worth pointing out that the knotting of the edges is exactly what determines the behavior when you cut the strip down the middle. Cutting a (single-half-twist) Möbius strip down the middle famously produces a single strip, because the edge is a single unknot. Cutting a double-half-twist strip down the middle produces two linked strips, because the boundary is the Hopf link. Cutting a three-half-twist strip produces a single strip tied in a trefoil knot.
I hope this collection of miscellanea contains something helpful. I suggest you look into the braid groups, because the braid group concept corresponds exactly to what you want to look at: what happens if you take $n=2$ separate strands and cross them exactly $k$ times before joining the ends together.
The monoid structure is useful, but there's not much reason to study it per se because the structure is completely known: it's a free commutative monoid with countably many generators. There is one generator per prime knot (where the knots are oriented for connect sums to be well-defined).
There are a few components to this result. The first is that prime decompositions exist (this follows from additivity of Seifert genus). The second is that connect sum is commutative (there's a straightforward isotopy). The third is that prime decompositions are unique (in Lickorish "An Introduction to Knot Theory," this is summarized in Theorem 2.12). This unique representation result implies that the commutative monoid is freely generated by the prime knots, so it's isomorphic to $\bigoplus_{i} \mathbb{N}$ where $i$ ranges over prime knot types.
A consequence of all this is that the data of an oriented knot is simply the vector $(c_1,c_2,\dots)$ with each $c_i\in \mathbb{N}$, all but finitely many nonzero, where $c_i$ counts the number of times prime knot $i$ appears in the knot. This vector is additive under connect sums, and every such vector corresponds to a knot.
Connect sums are a relatively natural operation on knots. From the point of view of the knot exterior, a connect sum is evidenced by an annulus that meets the knot along a pair of meridians. Or, from another point of view, if the knot exterior is viewed as a ball with a knotted tunnel bored out, then you're gluing the knot exteriors together along an annulus at a mouth of each tunnel, creating a larger ball and a larger tunnel.
[Edit: I'm misremembering something in this paragraph and the way I describe it doesn't work for non-split links. Leaving it in case someone knows how to fix it.] By the way, there's also a notion of a commutative semiring of links. When you do the connect sum of two links, you take all the pairwise connect sums between components of each link, and so the number of components is multiplied. The disjoint union of links is the addition operation and the connect sum is the multiplication operation. The empty link is the additive identity and the unknot is the multiplicative identity. This is isomorphic to the commutative semiring of polynomials over $\mathbb{N}$ in a countably infinite number of variables (one per prime knot type).
In any case, the concordance group for knots is a quotient of the knot monoid.
Best Answer
One ring I've thought a little about is formal linear combinations of oriented knots over some ring (for example $\mathbb{Z}$), with multiplication being connect sum extended by linearity. The existence of prime decompositions implies this is actually a polynomial ring over countably infinite generators, one per oriented prime knot. (So, for example, a generator for both the left-handed and the right-handed trefoil knots.)
The additive identity is $0$, the linear combination of no knots. Beware that $2\langle\mathrm{trefoil}\rangle$ is not $\langle \mathrm{trefoil}\mathbin{\#}\mathrm{trefoil}\rangle$; the latter is $\langle\mathrm{trefoil}\rangle^2$, which is different.
Knot invariants that are multiplicative under connect sum would be quotients of this ring. It is also possible to instead use disjoint union for the product and consider general links. If you take this ring over $\mathbb{Z}[t^{\pm 1/2}]$ and quotient it by a particular skein relation, the quotient space is generated by the empty diagram and the unknot, and the Jones polynomial of a link is the polynomial coefficient of the unknot representative.
This sort of ring is used by Chmutov and Duzhin for connected $3$-regular graphs, which in their case they are interested in graph invariants coming from Lie algebras so they take the quotient by the IHX relation.
Duzhin, S. V.; Kaishev, A. I.; Chmutov, S. V., The algebra of 3-graphs, Proc. Steklov Inst. Math. 221, 157-186 (1998); translation from Tr. Mat. Inst. Steklova. 221, 168-196 (1998). ZBL0944.57009.
Another algebraic structure, which I have no idea the utility of, is a semiring of links -- in the vein of what you describe. Addition is disjoint union, and multiplication is the disjoint union of the connect sum of each pair of components (in other words, we define multiplication on knots by connect sum and extend by "linearity" over disjoint union). The additive identity is the empty diagram, and the multiplicative identity is the unknot. This is an $\mathbb{N}$-algebra by having $n\in\mathbb{N}$ map to a disjoint union of $n$ copies of the unknot, which in other words means $n L$ is the disjoint union of $L$ with itself $n$ times. Though, if you formally add in the additive inverses, you do get a ring isomorphic to the first one I mentioned over $\mathbb{Z}$. This means the interpretation of $2\langle\mathrm{trefoil}\rangle$ would be $\langle\mathrm{trefoil}\amalg\mathrm{trefoil}\rangle$.