天保山 in Osaka is just called so. We can read it like てんぽやま, but this kanji-reading is not used. For example, any new mountain is found and given the same kanji name - 天保山 but てんぽやま？This is possible.
Well on the homepage of Osaka city they have the reading てんぽうざん so that's probably best indeed as OP suggested. https://www.city.osaka.lg.jp/kensetsu/page/0000011249.html Apparently the naming comes from the 天保 era (1831 to 1845) which is also read as てんぽう because at the start of it they wanted to make the Aji river deeper by dredging and the resulting mountain of sediment was then called 天保山. Quite funny actually, a mountain only a few meters high.
Anyway the reading info, as this is a proper noun, should come from the JMnedict database. And indeed there are some hits when searching for the string directly in the database web interface: https://www.edrdg.org/jmwsgi/srchres.py?s1=1&y1=2&t1=天保山&src=2&search=Search&svc=jmdict&sid=xxx These also can be viewed through Jisho.org https://jisho.org/search/天保山%20%23names It seems like the entry with id #5541003 got automatically matched with the corresponding entry data from Wikipedia which is why that one shows more prominently on the results page if you just make a regular search without the "#names" tag.
As for why JMnedict has duplicate entries here with different readings, I don't know. At least personally I don't see a reason. I will make a suggestion to the database maintainers to improve the corresponding entries and then they can have a look at it and make a decision. Ultimately this would then lead to the data on Jisho.org improving as well (as Jisho.org is mostly just a compilation of these dictionary files and some additional data from various sources as detailed in the footer and on the FAQ page).
天保山
Online I can mostly find the pronunciation てんぽうざん. Not sure if it's the only right pronunciation, but it should at least be an alternative reading, if not the only reading.