First, it's helpful to understand a bit about merchant republics. Republics are composed of Patricians, five cooperating families. Each one gets their own special palace holding and competes with each other and with other republics to build costal trade outposts. Adjacent Trade Posts that belong to the same Patrician family (thus also in the same Republic) merge together to form trade zones, which boost their own value (income and available loot), but also boost the tax revenues of city holdings. In other words, you are gaining more taxes from your costal cities that have trade posts in them. This revenue might be slightly higher if there were only one Republic with trade posts in your realm, but getting to that state would require destroying one Republic's posts and replacing them with another.
Embargo wars are full wars – there are no "cold" embargos, unlike the decision popup you received seemed to imply. You immediately become hostile to the enemy Doge or their Liege if they have one. If a popup doesn't tell you the whole story on teh CB, find the intended target and see what the conditions are if you were to declare the embargo war of your own volition: the conditions, rewards and penalties will be the same. The wiki seems to think that the requesting Doge should have joined your side. This may have been a one-time bug you experienced, or may have been something especially wonky triggered by being in the tutorial scenario. (The tutorial could use some improvement, especially with regards to the expansion features that have been added.)
Unlike other CBs, you gain warscore from sieging all of the trade posts in your realm. Since the trade posts in your realm are the real targets, it makes sense that you're the sole aggressor. The Doge who asked you doesn't have anything to do with your realm and probably doesn't have any of the opposing republics trade posts in their realm (their realm is probably filled up with their own trade posts).
Trade post destruction may or may not be sufficient to reach 100% warscoe by itself. If you lose battles, that will certainly set you back. If you white peace, you'll lose some prestige, but if you lose the war, you'll have to pay reparations, almost certainly more gold than you could have possibly won from loot. This will send you into debt if its more gold than you have, which is very bad.
Each trade post you siege down grants a fair chunk of gold depending on the level of the post and the value of the trade zone. If you win the war, you get a larger chunk of gold and all the trade posts in your realm belonging to that republic are destroyed. These can be rebuilt at great expense, so it's understandable that the controlling Doge will do everything he can to stop you. The requesting Doge is hoping to capitalize on your now depopulated realm to expand their own trade empire. But until someone rebuilds the trade posts in your realm, you'll be getting less tax revenue from your coastal cities than if you hadn't gone to war.
Also, Doge's typically don't have huge armies, but they frequently have piles of gold and freely spend it on mercenaries and gifts to allies to get them in the war. So either be prepared to defeat a few extra hosts of mercs, or run them around long enough that the Doge runs out of money (which may never happen, if their monthly income is high enough).
To sum up, embargo wars can be expensive for all sides. 150 gold isn't nearly enough to make it worth your while. Enter into trade wars with caution. The only time I've done an embargo war is when I had my own vassalized Merchant Republic who wanted to expand. This made it definitely worth it because:
- I knew I could take the target republic, their allies and all the mercs they could hire, making the gold earned from the war fairly attractive
- Most of the trade posts to be destroyed were actually in my vassals lands, who passed very little of their taxes up to me; I was actively trying to weaken a few of these vassals.
- I knew I would get vastly increased tax revenues from the vassal republic if they were able to take over all that territory.