This was originally posted as a comment on Reddit (now deleted) on September 3, 2019. I’ve made minor formatting changes and corrections, but otherwise haven’t done any editing or updating.
For future reference, I think the most barebones simple hydrating eye drops are probably Soft Santear (ソフトサンティア, pronounced Sofuto Santia; English PDF linked on that page). They supposedly have a similar composition to human tears, and are safe for contacts too. The only problem I have with them is that you have to use it up within 10 days after you open the bottle (which is why they come in tiny 5 ml bottles), because they don’t contain preservatives, and I personally don’t use eye drops very often.
There are also other simple lubricating eye drops without the menthol-y-ness or whitening properties. I’m guessing they misunderstood you somehow. It might help to say you’re looking for something for “dry eye” (ドライアイ, dorai-ai). I would guess most eye drops shouldn’t be used constantly regardless, though Soft Santear sounds pretty safe if you really must.
Also, if you wear contacts (コンタクトレンズ or just コンタクト, kontakuto) it’s important to mention that too, including whether they’re hard (ハード, hādo) or soft (ソフト, sofuto) lenses, since a lot of eye drops are not contact lens-safe. (Source: I live in Japan and wear soft lenses.)
There may be additional or different restrictions specific to other types of lenses, like colored lenses, one-day lenses, etc., but I don’t wear those and haven’t paid attention.
Personally the only time I use eye drops is when yellow dust or PM2.5 is bothering me. Rohto Alguard Contact a (ロートアルガード コンタクトa) helps with this.
I think people probably generally like the menthol-y sensation because it feels refreshing to them or it wakes them up more. The Alguard above also has a light version of this that helps temporarily relieve itchiness, so that’s another role it can play. But like the other better informed people have commented, it’s probably not a good idea to use these on a regular basis.
There is also usually some kind of indication of how menthol-y the product is on the packaging, in the form of the scale of 0–5 (or whatever number) as seen on the pages I linked to. I don’t know whether it’s universal, but in general 0 will mean no menthol-y sensation, and the higher the number, the more menthol-y it will feel (I don’t know whether it would necessarily be menthol in the ingredients, just using it to describe how it feels to me). That menthol-y sensation would generally be described as 清涼感 seiryōkan. Products with “cool” (クール, kūru) or 爽快 sōkai in the name/description would definitely have this property.
Original thread here, if you want to see what other users wrote.