If you want to Annex a Subject from another country in Victoria 3, you must declare an interest and later use the conquer state diplomatic play. You can declare an interest in another country's Subject by opening the Diplomacy menu tab on the left of your screen. After you have an interest, you can start Annexing another country's subjects in Victoria 3. However, if you want to annex your own subject or puppet, you must start an Annex Subject diplomatic play against them.
How to use the Annex Subject Diplomatic play
To use the Annex Subject Diplomatic play, open up the Diplomacy menu and select Diplomatic Plays. Under Diplomatic Plays, you can find the Annex Subject play at the far left of the options tab. If you have an interest or can Annex Subject, you should see a green number on the Annex Subject option. However, keep in mind using the Annex Subject Diplomatic play is a surefire way to start a war. Because of this, we recommend making sure you are not weaker than your subject, and they don't have any allies they can bring to the fight.
We recommend getting familiar with how to win wars and battles in Victoria 3. Getting familiar with the front system is also recommended to ensure you take your subject's lands when you decide war is unavoidable. But overall, as long as you have more troops, more military upgrades, and bigger allies than the Subject, you should be fine and will come out on top.
How to declare Interest in Victoria 3
To declare an Interest in Victoria 3, you must open the Diplomacy menu tab. Under the Diplomacy menu tab, you can find the option to declare an interest in a country close to you, as shown below. Click this option and select one of the countries closest to you that is a subject of a bigger power. Clicking the Subject with the interest option will declare your interest, allowing you to use the Annex Subject Diplomatic play later to start the annexation process.