The significance of kelvin as a unit of absolute temperature

You do not need to use the kelvins for the ideal gas formula. The important thing to know is that $PV \propto nT$ where $T$ is the absolute temperature and can be expressed in any arbitrary linear scale (as long as the temperature is still absolute), including the kelvin and the rankine scale.

However, if you choose the constant of proportionality to be $R$ given in J/(K mol), then the temperature's units must be in kelvins for the equation to make sense. But you're not forced to use such a constant. You could pick up any other constant and you could adjust the temperature scale you're using. In particular you could have picked the constant that would make the temperature in rankine units.

Edit: Since the title of the question refers to "formulas", let me mention a case I've seen up to several papers. With regard to thermoelectric materials, there is a $ZT$ factor that's used to gauge the quality of the material to be used as a thermoelectric one, for example to make a thermoelectric generator. The important thing is that $T$ must be the absolute temperature. Unfortunately many papers mention that $T$ is given in kelvins, while it is not necessarily the case. Fortunately enough, I would say most (good) papers refer to $T$ as being the absolute temperature and do not mention the units.

For example, =100, =8.314, =1, =8.314. Plugging in the values will result in =100 kelvin.

But what is the significance of 100 kelvin? Why could it not have been any other unit in an absolute temperature scale? e.g. 100 degrees Rankine

The answer is already contained in the first part of your question, when you write the units of the Gas Constant $R$. In order to get temperature in different units, one should use a different value and units for $R$. Similarly for a different thermometric scale.

Edit (improved the examples):

For example, in the case of Rankine degrees and using the same units for all other quantities, the perfect gas formula keeps the same form, but with a gas constant $R^{\prime}=4.61915$. In the case of temperature, $\theta$, in Celsius degrees, becomes: $$ PV = n\tilde R\theta+ Q n, $$ where $Q = 2.271 \cdot 10^3 = 8.314*273.15$ J mol$^{-1}$, and $\tilde R$ has the same numerical value of $R=8.314$, although now its units should be read as J mol$^{-1}$ $^{\circ}$C$^{-1}$.

There is a point that the other answers haven't made, which I believe is important: Celsius and Farenheit both allow temperatures below (what they call) zero. If you plug those measurements into the formula, you get a value of (pressure times value) which is negative; this is not physically possible. Note also that negativeness cannot be compensated-for by a simple constant of proportionality (R). Kelvin at least has the virtue of never being negative-valued.