Comparing acidic strengths between benzylammonium ion and phenol
The acid strength of each compound can be explained, but the acidity order is much more difficult to compare, because the two compounds are only remotely connected. It is misleading to conclude that the mere presence of a phenyl group somehow connects these molecules. The similarity of the pK$_a$s is likely a coincidence.
The question needs a clear definition of pK$_a$. We can ask what the pK$_a$ of phenol is, or the pK$_b$ of phenoxide. When we ask for the pK$_a$ of benzyl amine, we want the pK$_a$ of benzylammonium ion, not the pK$_a$ of benzyl amine going to C$_6$H$_5$CH$_2$NH$^-$. There is a bit of flexibility (or sloppiness) in the terminology here, and also a conflict in the values presented: the question quotes 9.33 for benzylammonium ion (I found 9.34 - no big deal), but the comment by the OP gives 8.82 for benzyl amine. The values should be identical because we know what the OP means. It appears that 8.82 is incorrect.
The difference between these molecules is far more important than any similarities. For example, one is a neutral molecule that you can put into a bottle with 6 x 10$^{23}$ more; the other is a charged piece of a molecule that you can hardly bring next to another one.
The phenyl ring has an enormous effect on the acidity of phenol, due to the resonance effect. It stabilizes the oxygen anion by more than 5 orders of magnitude, compared to, e.g., methanol (pK$_a$ = 15.5) or benzyl alcohol (15.4) or ethanol (15.9).
(Interesting that phenylethanol (pK$_a$ = 14.81) is more acidic than benzyl alcohol. This suggests that the methyl group in ethanol is more electron-donating than the benzyl group in benzyl alcohol; this agrees with the discussion in #3 below, for phenethylammonium ion.)
The phenyl ring has a much smaller effect on the amine molecule-ion; no resonance, just inductive. The pK$_a$ of benzylammonium ion (9.34) can be compared with, e.g.:

methylammonium ion (pK$_a$ = 10.66) Replacing the phenyl of benzylammonium ion with H reduces the K$_a$ by a factor of 21.

This is equivalent to saying that benzyl amine is less basic (pK$_b$ = 4.66) than methylamine (pK$_b$ = 3.34); therefore the protonated benzyl amine will give up an added proton more readily than methylamine. We can conclude that methyl is more electron-donating than benzyl.

ethylammonium ion (pK$_a$ = 10.71) Replacing the phenyl of benzylammonium ion with a methyl group reduces the K$_a$ by a factor of 23. Thus, ethylamine, like methylamine, is more basic than benzyl amine because methyl is more electron-donating than phenyl. (But we knew that.)

Phenethylammonium ion has a pK$_a$ = 9.73. Putting another CH$_2$ group between phenyl and nitrogen reduces K$_a$ by a factor of 2.5. This indicates that a benzyl group is less electron-donating than a methyl or ethyl group, in agreement with the comparison of the inductive effect between benzylammonium vs methyl- or ethylammonium ions and benzyl alcohol vs phenylethanol.


Benzyl amine and phenol are the neutral actors in this drama, and might react nicely in a one-to-one ratio. What would be the pH in solution? The graph shows the ratio of ion concentration to neutral molecule concentration for phenol and benzyl amine, separately, over the pH range of 5 to 9. They show contrasting slopes which cross at pH 7.33. At this pH, the ionic fraction of each compound is 0.21% of the whole - it doesn’t matter what concentration you started with if everything is soluble (benzyl amine is very soluble, but phenol, mp 43 C, is soluble in water slightly less than ~0.1 M in water at room temperature). Just add the same number of moles of each neutral compound to get 0.21% reaction and pH 7.33. What a coincidence! Practically the same pH as pure water.