Your "best friend and father of your son" can't also be your tax dependent unless you provide more than half their financial support, and they live in your home all year, and their taxable income is less than $4700.
(Your son can be your dependent, of course. If you live together unmarried with the father, then your son can be the dependent of either yourself or the father, but not both.)
Your partner might still be your tax dependent for 2023 if they still meet all 3 tests, even though they bought a car. If they don't meet the tests, they are not your tax dependent when you prepare your 2023 tax return. If you claimed 2 dependents on your W-4 when you started the job, you could give your job an updated W-4 that only claims 1 dependent (your son, if you plan to claim him).
Note that an adult dependent is an "other" dependent that gets you a $500 tax credit, so the effect of not claiming your partner for 2023 would be a $500 increase in your taxed owed or a $500 reduction in the refund you expect. Changing your withholding would lessen the effect of the change, with fewer dependents means having more tax withheld from each paycheck.
