I camp, not credit card tour. I try to keep up on laundry by doing sink laundry, thus try to minimize my clothing as much as I can but want to have enough that I am good for rainy periods where I can't hang laundry in the campsite. After a shower at a campground, I usually switch to normal clothes, my bike clothes might be hanging on my clothesline.
I am in USA, so social norms here are likely different where you are. I have also toured in Canada and Iceland.
What I bring is of course dependent on expected weather. My Iceland tour, I did not bring anything with short sleeves.
I like to have clothing that is bike specific, and other clothing that is "normal" clothing.
Bike specific stuff (jerseys and jackets and helmet cover are high visibility color):
Jerseys (2), might have long and short sleeve, might have both short, etc.
Bike shorts (1), mountain bike style with pockets. Usually have thin wallet in a zippered pocket and small multitool in pocket.
Bike pants (1) convertible to shorts with pockets.
Thin neck gaiter if it might get cold (near freezing).
Bike gloves (half finger).
Bike gloves (full finger), the ones I like also convert to mittens with a pocket over the fingers.
Rain cover for helmet. Also often wear in dry cool weather to keep the wind off top of head under helmet.
I have a long sleeve jersey that is full zip, thick enough to provide some insulation, I use that as a jacket when cool over other clothing.
Bike windbreaker.
SPD bike shoes. I have several pair, which I bring is in part weather related. (I use pedals that are SPD cleat on one side, platform on the other so I can use either pair of shoes on bike).
Rain covers for shoes.
Ear band for insulation over ears under helmet.
Both on and off bike:
Socks (4) that work both on and off bike with either footwear.
Most bikers I know do not wear underwear under their shorts, but I do, Exofficio brand briefs that are quick dry after washing. (4)
Rain jacket (1) that has hood, never use the hood on bike but nice to have in campsite.
Rain pants (1), long enough that they do not pull up above shoe covers when I pedal. I do not like ones that slide down, I use suspenders with them, I am probably the only one that uses suspenders for that.
Polartec vest (1), orange in case I want to use on bike.
Down vest (1), almost never used on bike but often used in campsite, yellow. If I expect it to be warm, might only bring one of these vests.
Off bike:
Thin stocking cap for cool weather.
Shirts (2), travel type so dry quickly. Might be long or short sleeve depending on where I go. I usually use shirts that have a zip pocket that is passport size.
Pants (1), nylon to dry quick, might be convertible to shorts.
Maybe shorts (1) if going some place warm.
Not clothing, but will mention anyway, three or four bandanas, might have a runny nose, use for hot pad when cooking, etc. Can be used for first aid. Cotton.
Shoes (1 pair), might be hiking shoes or trail runners. If I expect to go somewhere wet, Merrill Moab waterproof hiking shoes.
Often bring a pair of sandals to use as shower shoes or occasionally in campsite, but if you are indoors, that is probably not needed.
I am old enough that I need reading glasses for close up, have sunglasses and also a yellow pair of glasses that have reader inserts so I can see my GPS on the bike. Sometimes bring two pair sunglasses, gray for sun and brown tint for overcast, in addition to yellow for rain or fog.
Not clothing, but a rain cover for saddle is important if you have a leather saddle like I do.
Not clothing, but I think a rear view mirror is important, I like one on my helmet, a friend of mine prefers one mounted on a pair of glasses. Both of these allow you to see behind without taking your eyes off the road, but another friend of mine prefers one on end of is drop bars, but if you have one there you need to take your eyes off the road and might have luggage on the back that blocks view.
Did this from memory, probably forgot a few things.
Everybody is different, I am sure you will not find anyone else that has this list.
A friend of mine wears normal clothes on the bike on bike tours, part of that is that he does not want to bring any excess clothing and he thinks two sets of clothes is extravagant. He wears a high vis vest on his bike. He also avoids UV as much as possible due to skin cancer concerns, so he often wears long sleeves in hot weather.
I have used arm warmers and leg warmers, have given up on them. Instead I invested in some convertible bike pants and long sleeve jersey.
While I often avoid riding in rain near home, when on a bike tour that may not be as practical. Thus, I rarely wear my rain gear when home but often do when touring. So, keep that in mind when you shop for rain gear.
Have a great time.
If the three month tour would include camping, bring a clothesline, a flat drain stopper for sinks because campgrounds often do not have stoppers for sinks, etc. Even if all indoors, be ready to do sink laundry and have a plan for drying out clothing, such as drip dry in the shower, etc.