If I know a Sim who has a partner, but I want my Sim to be that Sim's partner, can I make them break up? For example, if I romance it hard enough, will an option like "ask to break up with partner" or similar become available? Or is it impossible?
Yes, it is possible to make other Sims break up. If you perform enough romantic actions with the Sim you are interested in, you will have the option of asking that Sim to break up with his or her partner, assuming the Sim in question is now a "Romantic Interest", as opposed to "Friend", "Good Friend" or "Best Friend". You will then be able to ask them to go steady, propose to them and eventually marry them.
Keep in mind that by asking a Sim to break up with their partner, you automatically make yourself an enemy of that now ex-partner.
YES. The old-fashioned way is to simply engage in romantic interactions while their partner watches or wait until a rumor hits about it.
If you have the University expansion, the Social Networking skill eventually unlocks the ability to freely alter social status between sims using your smart-phone app.
Supernatural can also break relations ships via love curses or certain potions similar to the old fashioned method by making sims fall in love with someone else which will eventually break the relationship.
If you have show-time, a genie can control a sim, temporary granting you the ability to use them as you see fit, and can simply call and break up with their partner, sell their house, vehicles, insult their children and friends, quit their job, and otherwise ruin their life.