My ex wife's boyfriend claimed my daughters on taxes for 2020....she wants me to claim them (I already filed my taxes and claimed one of my daughters like I usually do, she typically claims the other)

I believe you are saying that a man that is not related, claimed both of your kids on his tax return for 2020. In addition, you claimed one of your kids on your 2020. This means that both of you claimed the same child. The IRS will look at both returns and do an internal investigation.

• If the IRS decides that you should take the credit, you will get your refund. The IRS will send him a letter changing his refund.
• If the IRS can't decide, you will both get letters requesting documentation proving the child lived with you through medical records, school records, etc.

Once all of the original filing is settled and the money is handled, then an amended return can be file. If the other man wants to drop the kids, then you or your ex could claim the second. Unless, you ex files for the other child and goes through the above scenario. The IRS may decide he is not eligible for either after the incident above. This will be a wait and see event.