tl;dr
There is absolutely no contradiction. $I=S$ is purely definitional like saying all triangles have three angles. It holds always because we defined the word saving in such way to always be equal to investment same way as we defined word triangle as to always refer to object with 3 angles.
You seem to be committing equivocation fallacy by defining saving and investment in a layman terms (average joe on a street does not use saving and investment in their rigorous scientific definitions), instead of their proper economic definitions, and then wondering why economic models that use rigorous definitions do not seem to fit. The fallacy would be equivalent to wondering how come that in physics moon is satellite when (in layman's understanding) satellite is some man-made object in space and well people did not build moon.
If you use proper definitions there are no contradictions. Increase in interest rate actually has both negative and positive effect on savings (as it has both effect that lower and increase income and saving is properly defined as difference between net income and consumption). Due to way how saving is defined even if people put less money into financial assets (deposits at bank etc ) there is not necessarily less saving in the economy.
Full Answer:
It seems to me that this "S must equal I" is contradicted by the logic behind monetary policy. Interest rate policy seems to based on savings can be put aside somewhere and nothing is done with them. BUT S=I is the mantra of every economics textbook/model.
It is not contradicted by monetary policy.
First, you seem not to understand what $I=S$ means or how $I$ or $S$ is even defined in economics so we have to cover that first. $I=S$ is not mantra it is definition. That would be like saying in physics saying that shape of earth is oblate spheroid is a mantra, or saying that in geometry a shape that has 3 angles is called triangle is a mantra. Those are all definitions.
Likewise, $I=S$ is not some empirical observation it is consequence of us defining investment and saving in such way by construction. If you define word 'triangle' to mean any shape that has three angles then its not some sort of arcane mystery or magic that anything with three angles will be labeled triangle.
Furthermore, this definition is only valid when government runs balanced budget in closed economy. The $I=S$ comes from definition of GDP in closed economy when government runs balanced budget. By definition GDP in closed economy is given by:
$$Y = C+ I +G$$
Where $Y$ is output/income, $C$ private consumption, $I$ investment and $G$ public consumption. By definition saving is $S=(Y-T)-C$, where $T$ are taxes, hence saving is not simply putting money in bank. Any time you do not devote your income to consumption of final goods and services you are saving under the definition above. What even more any time government taxes you, the economy is actually increasing public savings. Using both definition of $Y$ and $S$ (by substituting the definition of $S$ into definition of $Y$ by solving for $Y$) we arrive at:
$$S + T + C = C + I +G \implies S +G -T = I $$
Now if we assume balanced budget and hence no public saving we will have $G=T \implies S=I$. This purely definitional like saying that triangles have three angles, that degrees in circle sum up to 360 degrees etc. It is not empirical observation, its just a definitional. Note the terms are again not even defined as regular people understand them. Investment for example includes inventory investment. Hence any time firm is not selling some goods that would be considered investing. That is not the layman person definition of investment but all sciences have specialized language (e.g. layman would not call moon satellite yet that's astrophysics definition).
Hence we have:
$S = I$ or in words investment equals private saving, in closed economy with balanced budget.
$S + T-G= I $ or investment equals private plus public saving, in closed economy without balanced budget.
$S + T-G +NX = I $ or saving equals private plus public plus saving by accumulation of trade surplus, in open economy without balanced budget.
We always have different kinds of saving on LHS and investment on RHS so some textbooks will just state saving equals investment, however that is not the same as $S=I$ which would be only private saving equals investment.
Second, you are not completely correct in saying:
But when it comes to monetary policy, central banks increase interest rates in order to slow inflation (and in doing so slow investment). But higher interest rates will increase the attractiveness of saving thus increasing investment (S=I)
A basic textbook framework for understanding monetary policy is IS-LM model (I will use the closed economy version to save time). In IS-LM model we have the IS (interest saving relation):
$$Y = C(Y-T) + I(Y,i) + G$$
Which is the same formula as definition for GDP just with emphasis that $C$, and $I$ is also a function of other parameters like $Y-T$, $Y$ or $i$. The whole reason why this is called interest saving relation is that $Y = C(Y-T) + I(Y,i) + G \implies S +G -T = I$, as already proved above. Again this is just definitional like that triangle have 3 angles or that shape ball has is sphere etc. This is not fundamental truth or empirical observation, it is just a consequence of people who first came up with these definitions like Keynes and others, defining saving, investment etc in such ways.
Next we have the LM (liquidity-money) relation given by:
$$M/P = L(Y,i)$$
where $M$ is money supply, $P$ price level (increase in which gives you inflation) and $L$ which is money demand that depends on $Y$ and $i$.
Now to further solve the model we have to concretely define the functional form of money demand $L(Y,i)$ consumption $C(Y-T)$, and investment $I(Y,i)$, for the sake of simplicity I will use basic linear functions for all of them, the exact functional form does not matter but mathematically linear functions are easy to solve. Hence let's say consumption is given by:
$$C = c_0 + c_1(Y-T)$$
Where $c_0$ is autonomous consumption and $c_1$ marginal propensity to consume.
Investment is given by:
$$I = I_0 + d_1 Y - d_2 i$$
where $I_0$ is autonomous investment and $d_1$ and $d_2$ are effects of income and interest rate.
Furthermore, note this also implies that $S + T -G= I_0 + d_1 Y - d_2 i$ so it is not true that low interest rate discourages saving directly. However, note here saving does not mean putting money inside your bank deposit, saving is difference between income and consumption, lower investment lowers income so it has to lower also saving because you have less income from which you can save money.
Higher interest rate does encourage people to put money inside bank or other financial assets as opposed to saving by not consuming your resources in form of improving your business or other forms of not consuming your resources (which is how saving is actually defined). For example, higher interest rates would affect negatively your saving in form of forming inventory etc. Now as we will see later lower interest rate also encourage economic activity by stimulating $Y$ and that also increases saving (as defined in macro), so there are both positive and negative effects of $i$ on saving and simply stating that $i$ encourages saving, as defined in macro, is not correct (not in general equilibrium model anyway, sure if you just have partial equilibrium of financial market (e.g. market for bank deposits) then higher interest rate means more supply of saving, but that's only one form of saving not all saving in the macroeconomy.
Next, the money demand can be defined as:
$$L = f_1 Y - f_2 i$$
using the definitions above we have:
$$\text{IS: } \quad Y = c_0 + c_1(Y-T) + I_0 + d_1 Y - d_2 i + G \implies Y = \frac{1}{1-c_1-d_1} \left( c_0 + I_0 +G -c_1T \right)- \frac{1}{1-c_1-d_2} i $$
$$\text{LM: } \quad M/P = f_1 Y - f_2 i \implies Y= \frac{1}{f_1} \frac{M}{P} + \frac{f_2}{f_1} i$$
So as we can see above interest rate has both negative and positive effect on output and thus also on level of saving. Changes in interest rates certainly can change the relative share of $S$, $G$, $T$ or $I$ in the economy but the identity $S +G -T = I$ will still perfectly hold. Moreover, if we subdivide $S$ into saving in form of deposits $S_1$, saving in form of accumulating more inventory $S_2$ etc, we could again see changes in interest rate changing composition of $S$ between $S_1$ and $S_2$, but the overall identity $S+G-T = I$ is left unchanged. If we assume that government runs balanced budget then we can make the same claims about $S=I$. Again, interest rate increase may encourage saving in some form (e.g. deposits) but it discourages saving in other forms (e.g. accumulating inventory, putting money under mattress).
