Please explain how you filed your tax return. Did you and your spouse file a joint return with both names and both Social Security numbers on the return? What do you mean when you say you paid "on my wife's SSN?"
On your own, there's no way to do this. it would seem your options are to
1) contact the IRS
First choose your language.
When asked for SSN do nothing it will ask twice
You may then be transferred to an agent, but there is likely to be a long wait. or you might not even get through. the IRS is receiving more calls then it can handle so may do not get through and other get disconected
2) if you got a notice, first try calling the number on it. if no number or you can't get through send it back with a complete explanation. sign and date. provide both SSNs and print your names. the IRS is far behind in processing paper forms. it could be 4 to 6 months before the IRS gets to this. in the meantime you may get a notice of levy for failure to pay. that notice will provide means to protest.
3) contact the IRS Tax Payers Advocate Service in your state
4) see if you can contact a local IRS office near you and set up an appointment.
I'm not 100% certain what happened. I think we had my name as the first name on the tax filing. However, I think the payment was made under my wife's name.
I presume that if we can't merge the profiles to identify us as married-filing-jointly, the only way to remove the threat of a collections from my login is to pay the same balance, merge the households, then presume we show a credit rather than a $0 balance.
"How do I merge my SSN and my wife's SSN to reflect a single household"
File a joint tax return : filing status MFJ.
If that didn't happen you can use 2020 Form 1040-X to do it (change your filing status).
You will report both incomes and all amounts already paid.