I keep seeing $\Bbb R/\Bbb Z\cong S^1$. Shouldn't this instead be the quotient of $\Bbb Z$-many copies of $S^1$ by some fixed point? Isn't the correct quotient $S^1\cong\Bbb R/\{\{x,x+1\}:x\in\Bbb R\}\cong [0,1]/\{0,1\}$? Is it an abuse of notation? Am I going mad?
My thinking is this:
Fix $0\in S^1$ (we'll think of this as the "origin," but it doesn't actually matter what point we choose). Then,
$$\Bbb R/\Bbb Z\cong\left(\coprod_{n\in\Bbb Z}S^1\right)/\{(n,0):n\in\Bbb Z\}.$$
"proof": Let $n\in\Bbb Z$. Then $[n,n+1]$, considered as a subspace of $\Bbb R/\Bbb Z$ is a circle - it is locally homeomorphic to $\Bbb R$, and $[n]=[n+1]$. Choose $x\in\Bbb R,x\notin\Bbb Z$. Clearly $[x]\ne[x+1]$, so the circle $[m,m+1]$ is distinguishable from $[n,n+1]$ at all points except $[0]$. Tag each interval $[n,n+1]$ with its left (or right) endpoint, and we have
$$\Bbb R/\Bbb Z\cong\{(n,x):n\in\Bbb Z,x\in[0,1]/\{0,1\}\}\cong\left(\coprod_{n\in\Bbb Z}S^1\right)/\{(n,0):n\in\Bbb Z\}$$
It's an angry atomic orbital! A bunch of balloons! A big ol' flower!
It isn't a circle.
Clarification on notation: I didn't realize that it isn't a universal, but I've always thought of
$$\coprod_{i\in I}X_i=\{(i,x):i\in I,x\in X_i\}$$
and
$$X/Y=X/(Y\times Y\cup\operatorname{id}_X),$$
where $Y\times Y$ is taken to be the equivalence relation equating all points of $Y$, as standard.
Also, I just realized $\operatorname{id_X}=(=\cap (X\times X))$, and that's wonderful.
For better or worse, there is a standard abuse of notation in this context: when a group $G$ has an "obvious" action on a set $X$, we abbreviate the quotient set $X/\sim_G$ (where $a\sim_G b$ iff $\exists g\in G(ga=b)$ via the above-mentioned action) by $$X/G.$$ This is usually not confusing, but when the group is also a subset of $X$ it can be - e.g. in the case you mention of $\mathbb{R}/\mathbb{Z}$, which one may reasonably mistake for a "bouquet of circles" sharing a common point. But here we're thinking of $\mathbb{Z}$ as a group (acting on $\mathbb{R}$ via the shift action $(z,x)\mapsto x+z$), not a subset of the "starting space" $\mathbb{R}$.
For that matter, your interpretation is also an (extremely common) abuse of notation, since it conflates $Y$ with $Y\times Y$. The only truly unambiguous notation for quotients is $X/E$ where $E$ is literally an equivalence relation on $X$ (and even here we have to make an agreement to not interpret things differently - it's possible for an equivalence relation on a set to also be a subset of that same set, after all!).