Manuals are getting rarer and rarer, even in europe these days automatics have become a lot more common (many of them are kind of a manual-automatic hybrid, where you can shift the car using paddle shifters, or allow it to work in full automatic mode, no clutch). That said, in many places outside the US if you rent a car, it could be a manual.
I miss driving a manual in many ways, when my wife made me get rid of my nice subaru wagon (she called it disreputable), I cried. There are advantages to manuals over automatics, even with the CVT automatics and with automatics with computer controlled shift points, manuals get better gas mileage. On bad road surfaces, with a manual you can slow a car down without using the brakes, using sequential downshifting, in general with a manual you have more control over the car. It isn’t as great as it once was, back when your typical automatic was a land yacht, the differential was greater, these days with the automatics in cars, and with the improved suspensions and steering and AWD, it probably matters a lot less.
I don’t know of any rental car agencies that rent manuals, other than some types that rent sports cars or classics. It is very hard to find a manual these days, they don’t offer it on a lot of models any more (I find it amusing that when you look at the sticker price of a car, they still consider an automatic transmision an 'extra", given it is standard on about 90% of cars these days.
I suspect you will find you never really lose the ability, I drove a manual the other day and it was no big deal, even though I haven’t driven a manual in a while. One of the things with manuals is I drove a wide range of cars, from econo buckets to high powered muscle cars and sports cars, and they all drive differently, so it gets ingrained on how to approach driving a manual transmission car. Personally I think all kids should be forced to learn to drive a manual, if only to understand a bit of what is going on when they drive it, so it isn’t turn the key (push the button) and go, same way I wish they would teach kids the dynamics of driving and how a car is put together, so they will realize what can go wrong and be a little more careful. I realize that as time goes on, if auto driving cars become the norm, it won’t matter any more, but until then, would be great if they did know.
As far as teaching someone to drive a manual, I grew up on manual cars (we had a couple of Alfa spyders and a manual transmision wagon), we didn’t have automatics (friend of mine claims I was the only person he knew who could stall an automatic). Anyway, when my wife and I were dating, I taught her to drive a manual on one of the Alfas, that had one of the trickiest clutches to use I have ever driven, it engaged roughly and you had to feed it the gas just so. She claimed that I was trying to torture her and secretly wanted us to break up so did that to get her to do the breaking up, until she realized that people who own Alfas torture themselves:)