I never got rid of my pickup truck, but I have owned a few small displacement two-wheelers that I used for commuting year-round for several years in my 20s. Mind you I live in North Carolina, so A. I'm going to use miles per hour and miles per gallon, B. the climate here is warmer year round than almost all of Europe, and C. when I talk about any legalities like licensing, that's what I'm talking about.
I owned a Yamaha Razz (an ancient 50cc 2 stroke scooter), a modified Aprilia RS50 (rare bike to find in the US, by the time I got it this is a 75cc 2 stroke motorcycle, 6-speed manual transmission) and a Kawasaki Ninja 250F (250cc 4 stroke 2 cylinder, 6 speed manual). The scooter I mostly stuck to roads with speed limits of 45mph or under, and still that was pretty sketchy as in most cases I could not keep up with traffic. The other two bikes I've driven all over creation on every kind of highway we have.
Performance
The Aprilias 75cc two-stroke was just barely capable of pushing the bike to about 80 mph; it is my understanding that the RS50 is often limited to 30mph in the UK, and wide open it could do about 50. A two-stroke engine with nearly no torque isn't great for stop and go traffic; this bike was best on the backroads where you would stay at 45 to 55 mph. It made around 60 miles to the gallon, it did consume oil, wasn't that easy to start in the winter, etc. Cruising on divided highways and interstates was asking a lot of an engine that size. I did occasionally carry a passenger, which was a LOT to ask of that bike.
The Ninja 250 is a perfectly adequate machine for highway use. The chassis wasn't quite as sporty as the Aprilia, with a more upright riding posture, which just made it more comfortable. It was easily capable of over 100mph, it held nearly 5 gallons of gas and made 70 mpg, so it had tremendous range. I still haven't found a motorcycle that can do what that bike did. It was easy to handle in traffic and quite capable on the freeway. Carrying a passenger on this bike could be a challenge with any cargo; it was kind of a choice between saddlebags or a passenger, not for weight but for space.
I would estimate that a 125cc four-stroke with a CVT would perform about like the Aprilia or worse, probably more torque but less horsepower, you're probably locking yourself out of long distance commute by major divided highway.
Practicality
I only really started doing things like grocery shopping by bike when I got the Ninja. I had both a tank bag and saddle bags for that bike, which were frequently used. If I could do it again I would do permanently mounted and weatherproof saddle boxes. Some scooters come with trunk space under the seat but others are full of fuel tank since they often keep the space in front of the driver empty.
Weather
I rode bikes through 100 degree summers and in freezing rain. None of my bikes were really equipped for shielding the rider from the weather. The coldest I've ever been was in the saddle of that Ninja. If you don't like being rained on, commuting by bike isn't for you.
Riding gear is essential. I would budget about $1000 USD (about the same in euro) for a helmet, jacket, boots, pants, gloves and rain gear. I actually wore a jacket that was designed for summer weather, a nylon mesh jacket that breathed very easily, year round. IT had an internal rain liner I could put in which warmed it up a little, and I wore a fleece under that in winter, and still froze my ass off. For most of the year I didn't bother with the armored pants I bought because they were cumbersome; for commuting you might look into "steel jeans" or similar. Your gear is there to keep you comfortable when moving at speed and to keep you in one piece while crashing. Which you WILL eventually do.
Maintenance
Bikes need more maintenance than cars. Tires don't last as long, especially the tiny ones thy put on scooters. Those smaller, simpler engines are fussier. They might take less oil, but they often take expensive synthetic oil specifically designed for motorcycles, and it requires changing more frequently.
Everyone in cars is trying to kill you
No seriously everyone who drives a car including you and me are ignorant and malicious dipshits when it comes to the safety of those on two wheels. You will be pulled out in front of, you will be swerved in front of, you will be merged into. Much more so than in a car you will have to defend yourself. Keep your horn working, keep your eyes working, learn karate, install a Phalanx CWIS system if it's within your budget.