$$\mathrm{Pr}(\mathrm{coal} \, | \, i=2) = \frac{1}{N}$$
Probability of first lineage picking an arbitrary parent is 1, while the probability of the 2nd lineage picking the same parent is $\frac{1}{N}$.
Probability of coalescence scales inversely with population size.
$$\mathrm{Pr}(\mathrm{coal}) = \binom{i}{2} \frac{1}{N} = \frac{i(i-1)}{2N}$$
There are $\binom{i}{2}$ ways pairs of lineages can pick the same parent.
Probability of coalescence scales quadratically with lineage count.
$$\mathrm{E}[T_i] = \frac{2N}{i(i-1)}$$
This is a geometric distribution. If each generation there is a $\frac{1}{x}$ probability of an event occurring, we expect to wait $x$ generations for the event to occur.
With per-generation probability of an event $\frac{1}{x}$ small, but many generations, then the
discrete time geometric distribution approximates to a continuous time
exponential distribution.
Thus, we assume $T_i$ to be exponentially distributed with mean
$$\mathrm{E}[T_i] = \frac{2N}{i(i-1)}.$$
These approaches directly estimate the pairwise rate of coalescence $\lambda$, which is measured in terms of events per year. Thus, the timescale of coalescence $\frac{1}{\lambda}$ is measured as the expected waiting time in years for two lineages to find a common ancestor.
The timescale of coalescence $\frac{1}{\lambda}$ is equal to $N_e\tau$, where $N_e$ is measured in generations and $\tau$ is measured in years per generation. $\tau$ acts to rescale time from generations to years.