m = n log_n(m)
Because m and n are distinct (ie: m!=n), let's say that m>n.
The log term will always be greater than one. Imagining the shape of the log function (with base>1), it isn't hard to see that there is one solution.
Not a rigorous proof, but neither was the article.
The mistake in your proof is that you considered the shape of the log function while implicitly holding m on the left-hand-side constant (i.e. you should have concluded "for a constant m", there is only 1 n satisfying this equality"). However, since m is variable, you have to consider an infinite number of log functions. If I missed something, let me know.
Would be interesting to use a recurrence relation by substituting m