There are a few things to consider when it comes to claiming dependents if you're an unmarried couple. The first is that you must be able to show that you provide more than half of the dependent's financial support. This can be done by providing documentation of your expenses, such as rent or mortgage payments, utility bills, food costs, etc.
If you meet this criterion, then either person in the couple can claim the dependent on their taxes. It's generally advisable to have just one person claim the dependent in order to get the maximum tax benefit, but there may be circumstances where it makes more sense for both people in the couple to claim partial dependency (for example, if one person has a much higher income).
It's also important to note that being married does not necessarily mean that you can automatically claim a dependent. The same rules apply - you must provide more than half of the financial support for the dependent in order for either spouse to be able to claim them on their taxes.