What are the advantages of a custom frame for a touring bike?

You're almost certainly better off spending that money on going touring.

I'd go so far as to say that until you have done enough touring to have a list of things that you want from your touring bike that you can't get from any production bike, don't bother thinking about it. Well, unless your other hobby is spending money, in which case I suggest buying one from each custom builder and seeing how they compare.

The main advantage of a custom frame is that it's built to fit you and has exactly what you want on it. Long term tourists often develop quite specific requirements and no mass-produced bike will satisfy them. Even simple things like built-in racks are depressingly rare in mass-market touring bikes, let alone more specific things like eyelets for tying things to. But again, unless you already have that list of things you're wasting your money getting a custom bike.

The main risk is that you spend a lot of money then decide that it wasn't what you wanted after all. This is especially the case with people who haven't done a lot of touring, and I know at least one builder who refuses commissions from novices for this reason. Every unhappy customer they've had has been a newbie to whatever they wanted the bike for. The solution is to put them on a cheap bike and wait a year.

With S&S couplers you're likely to need those added after you buy a mass-market bike, which is an opportunity for you to get the framebuilder who does the work to add any little extras you want. It'll very likely be cheaper to buy a new mass-produced bike and have the couplers added than get a customer bike. But it's likely to be cheaper to pay a freight company to ship the bike around the world a couple of times than to get the couplers fitted.

As Alan says, you can tour on any bike. I've done a 5Mm trip through northern Australia with people who rode BSOs (bicycle shaped objects), one of whom did the last couple of megametres with no balls in their bottom bracket (and hence, no front derailleur). They didn't enjoy the riding as much as they could have, but they did it.

As far as comfort goes, you will very likely be more comfortable on the cheapest recumbent you can find than the most expensive upright bike.

Before you decide on a custom frame or coupled bike you should have a very good of idea of what type of touring and how far from home you intend to go. I say this as someone who has made close to 100 multi-day tours since the mid 1980s, none over 14 days long with the average around 7 days. I started with a mid-80s Trek 520 (very different from the current bike of the same name and purpose) and now close to dozen touring/travel bikes later have a Salsa Vaya Travel. Different bikes are appropriate for different with fit and saddle the best measure of comfort. If you drive to your tour start/end point S+S coupling (or folding bike collectively 'suitcase bikes') are not priorities or money well spent. Start with event tours (the sponsor carries your gear from camp site to camp site) on a second hand touring or cycle cross frame that fits well and make decisions based upon your experience.

Points to consider:

1. A touring specific general bike is probably better, vs. the custom frame of a generalist shop. Geometry, strength, details (braze ons, eyelets, racks, etc.). If it is a not from a touring expert, your bike can be actually worse for touring vs. a touring expert's general bike.
2. Is it a custom sized frame of an otherwise stock bike, or is it built from ground up? The latter is a good way to go, for the latter, you need a couple of tours behind you to tell what you need.
3. The comfort on your bike is primarily determined by the contact points (unless the bike is way too small or way too big): handlebar, saddle, pedals. For example, my experience shows that gel tape improves more hand comfort (30 USD) on touring vs. suspension forks (300 USD).

My advice:

If you have the budget, make a custom built frame. It's expensive, and the price/performance ratio is low (the benefit you get is probably marginal vs. the investment), but why not. It will be your bike, made for you, it's a great feeling, and people easily spend more money to upgrade the stereo in their car.

However, the custom frame has to come from a touring expert workshop, who also manufacture several different stock touring bikes. So the basics about geometry (chainstay, wheelbase, fork angle) are fit for touring. Koga, IDworx, Tout-terrain are good examples.

My own experience is that the more I tour, the less I care about the bike, and focus more on the journey. At the beginning, I worried a lot, now, I just use the bike I have. It has 8 hub gears, but it doesn't stop me to go for several days of touring, even with hills.