Wood – What kind of wooden fence is this and how to go about staining it

Ok let's talk outdoor staining. You are right that the first thing you need to do is clean off any mold, dirt and green slime. As with decks, a good and cheap solution of TSP, bleach and water is a fine cleaner, no need for expensive premixed stuff. Using a one gallon garden sprayer, apply your TSP/bleach solution. Let it sit a bit, but not dry. Scrub with a stiff brush and rinse with a pressure washer or high pressure hose. The drying time is obviously tricky in your area, but usually a couple of days of decent weather does a good job. You can somewhat judge the dryness by it's color as compared to the color when wet.

Since the last stain you used was a urethane based acrylic, you really need to stay with an acrylic product this time. A pure oil based product may not sit well and raise the acrylic and create a mess. Also, since there was spar urethane in the last application, a stain only may not want to stick to the areas where the urethane is still in good condition. The urethane is a hard resin and will repel a simple stain. I would probably look towards using a similar product to what you used last time.

As far as what brand to buy, don't over think it. You are painting a fence, not a church! Avoid any bargain basement crap that is most likely low quality. I love Sekins, but it is very pricey. Most major brands are good. My advise is to try to use the same Behr you used last time. It will be formula compatible and the color will match. Good luck

Yes, talk to your neighbor, but you can just do your side. I've pressure washed my fence that looked like that (entire fence...I wasn't matching another part of my fence,) and removed the mold (that's what has turned the boards gray) down to where the boards are a golden tan again. Try doing just your side of the fence without hitting the edges of the boards and I think you'll get 99% of the effect. Then, match up some stain to the existing stained boards and it'll match fine. It looks like the existing stained fence could use a fresh coat too. Remember, the same stain "recipe" for the new fence will probably not be the same "recipe" for the old fence. Always try a small sample in an inconspicuous location.