Oxidation of naphthalene with KMnO4
I already made a comment about some of what I am about to say but I will provide a partial answer. I say partially because I could not find any mechanism for the second product. However, from literature, what I found was that in acidic conditions, $\ce{KMnO4}$ will oxidize naphthalene into the first product, phthalic acid1. In basic/alkaline conditions, $\ce{KMnO4}$ will oxidize naphthalene into Phthalonic Acid2. The results also agree from other papers. The proposed mechanism for the acidic solution reaction is here (taken from the paper in the first link):

It is essentially the same mechanism for the cleavage of a double bond but it happens in two positions in naphthalene. Note this is the mechanism for $\ce{KMnO4}$ oxidative cleavage:

Also note that the mechanism in the first image was based on a 2017 paper studying the kinetics of the reaction and then formulating a mechanism. So this reaction is still being studied. Along the same lines, I could not find the mechanism for the reaction of $\ce{KMnO4}$ and naphthalene in alkaline conditions. I would assume that its mechanism is still being researched.
References

https://www.jchps.com/issues/Volume%2010_Issue%201/40-0581116.pdf
The Oxidation of Naphthalene to Phthalonic Acid by Alkaline Solutions of Permanganate, R. Arthur Daly, The Journal of Physical Chemistry 1907 11 (2), 93-106, DOI: 10.1021/j150083a001
http://www.orgsyn.org/demo.aspx?prep=cv2p0523
https://www.acros.com/_rainbow/pdf/oxidation_brochure_manga.pdf
http://studymaterial.unipune.ac.in:8080/jspui/bitstream/123456789/8059/1/POLYNUCLEAR%20HYDROCARBONS.pdf