Hi,
I'm pretty new here, I was wondering if someone can explain how this artifact works. I checked past threads and the consensus seems to be that it gives a certain x% of money made from territories, not territory bonuses. I calculated it multiple times by adding money from territories alone, but still the numbers don't match. What am I missing?

The problem is that you can't see your territory income as a "money per second" number. Even with the respective Statistics advancement level, you will only see "money from territories so far accumulated". Therefore, it is very difficult to check these numbers on your own. But to me it always seems plausible, so I would trust the numbers in general.

You are right that your money income over time (therefore, excluding everything that happens only occasionally, like caches sells, clan requests) consists of bonus income and territory income. The former is the circles, squares, sometimes hexagons with numbers on the map, the latter is noted on the information window for each territory individually. A good chunk of territories provide some kind of income, but some don't and in general bonuses give more. Therefore, for a territory income boost artifact to be good it needs a significantly higher percentage than a bonus income boost. If you have both, just swap one for the other to see whether your overall money income goes up or down, or just use both if you can.

Let's say, you have 23k/sec from territories and 42k/sec from bonuses. Your overall money income would be 63k/sec (and this is the only number you can see in the money dialog, when hovering your money balance). Say, you had a 50% territory income boost, your territory income would be 34.5k/sec now, and the overall income would be 76.5k/sec. The "problem" with this is, that there are also other ways to improve your money income. There are techs to boost both territory and bonus income and there is at least one advancement for bonuses. For example, if you have 25% territory income boost already and gain 50% on top, that won't be 1.5 time the already 1.25, instead you will go from 1.25 to 1.75. I'm short in time right now, perhaps someone else can explain that of I will when I come back.

Hope that already helped a bit.

Oh no you have been very thorough, thank you!
I haven't though about how multiple bonus would work because I have none at the moment, but if I understood correctly, basically they would all apply on the base value, instead that on top of each others, right?

yes, the total income is:
base*(1+bonus1+bonus2+bonus3+etc)
so you end up with less armies than if would multiply the bonuses

So to be clear ...

total money income = territory income + bonus income

Where:
territory income = territory base income*(1+buff1+buff2+buff3+etc)

bonus income = bonus base income*(1+buff1+buff2+buff3+etc)

(renamed bonus1/2/3 to buff 1/2/3 to avoid the buff/boost concept getting confused with the 'bonus' income concept itself)

What I really missed (everything else seems to be mention by someone by now) is that there is the "Modifier" dialog that you can unlock using the Statistics advancement (Level 3 or 4, not sure right now) that appears on the "Army" tab and lists all active modifiers and also how they combine. That way you will see the actual effect of several boosters on the same income.