I recently purchased a rototiller, which as you of course know has a similar gear box as a mower, driven by PTO. My rototiller actually calls for 85w 140 in the manual, in both the pto gear box and the larger side gear box. I think you will be fine. The oil is a little "thicker" than 80w 90, and will handle heat better. many 85w 140 gear oils also have "EP" or "extreme pressure" additives. a lot of 80w90 gera oils have "L/S" or "limited slip" additives. So the real differences as I see it are as follows:
80-90 gear oil is not as thick, so it is more fluid. (only slightly though - hard to tell difference dipping your finger in it) because of its lighter/thinner weight it isn't as good in high heat or perhaps in areas where the gears in a box are experiencing a lot of stress. This is likely why my rototiller calls for 85w140 and your mower calls for 80w90. Rototillers tend to hit things like rocks and can encounter a lot of resistance and knocks, causing this to transfer some of that force into the gears of the unit, so they recommend a heavier oil that is better under heat and extreme pressure. Mowers by contrast shouldn't be encountering the same resistance and "jolts" to the rotational system, so less heat, less stress, and so they recommend a slightly lighter, more slippery, more fluid oil. I have seen some rototillers, especially older ones, that did call for 80w90 oil however, so really the differences in these oils is likely fairly small.
So really gear oil is gear oil. One is just a tad heavier, tad thicker, tad better under heat and extreme pressure. The other is a little lighter, little more fluid, little more "slippery". They could largely be interchangeable in rototiller / mower gear boxes. If you want to play it super safe and are concerned about the oil, then you know the main concern with using this oil over the recommended oil is that the oil you have in there is heavier. Let your mower idle a tad before using it. Put it up to 540 pto and let it run for a minute before mowing. May not help much, but the idea is of course to let that slightly thicker oil warm up a tad, so it is moving better, and get the gears nice and coated, before dropping it down and go to cutting.