There are plenty of these seats available on Amazon and therefore should ship in a plain Amazon box. ordering one and getting it delivered would have the same advice as buying diapers online and getting them delivered, which has been discussed a lot already, things like waiting for an "empty house" day and paying for next day delivery, shipping to a collection point etc.
As for advice on the rest, they are generally quite thin and low profile so fairly easy to hide in a drawer or the bottom of a cupboard/closet, behind a desk etc. the hardest part really would be getting it in and out of the toilet without someone coming past as you move between, the easiest option there may be to hide it under something (like a towel).
It may seem a little odd but for instance I keep a towel in my room for days when it's pouring with rain or so hot I'm sweating since I sit on my bed when watching tv/playing games so my excuse for it is I don't want my bed getting wet from rain/sweat.
Depending on how busy the house is, how far you have to travel room to room and how loud people are moving around, it may not be too hard to just wait until no one is on the landing and just carry it. For back up try to find a spot in the bathroom that it can be stashed quickly if someone else is waiting outside to use the toilet (for example I have a laundry basket in the bathroom so I would be able to just throw it in there under some clothes), you can then collect and move it once they are done.
As for cleaning it's not that bad, sure there are germs on and around a toilet but unless you've peed or got poo on it then it probably doesn't need wiping every time.
When using for children you might clean it every time but that's more because children's immune systems are weaker and more susceptible, they also are more likely to get pee/poo on it and to be putting their hands all over it to get on and off and then even if they wash them (which they often don't do properly) putting their hands in their mouths.
When it does come to cleaning a quick spray with general purpose antibac and wipe with a tissue is fine. If that may not be possible secretly in a your bathroom you can buy antibac wet wipes and keep them in your room to just wipe it down.
The other piece of advice I'd have for this is to be careful what you buy, these are designed to make the toilet seat smaller for toddlers bottoms so while the hole might be big enough for you to go in, the seat itself might be too small for you to fit or the hole will be in a position where peeing is fine but pooping with hit the back (or vice versa).
They often have high back, sides for a toddler to hold onto and help stop them falling off and a splash guard in front, fitting an adult or older teen bottom is not possible the answer to this would be looking for one with a very low profile like this
Rather than ones like this to be sure you can sit on it comfortably.
You can also look for "toilet seat raisers" which are designed for older special needs children and the elderly who may need extra height to be able to get back up. these are much taller and would be harder to hide but are designed for bigger bottoms so they might fit you better.
If the slimmer seats and raisers are not to your liking due to lack of pictures or theme designs you can always buy a pack of stickers for your favourite thing (i.e. paw patrol or Marvel) to put on them.