Re: titles.
They can't without royal permission.
Well they could, but if they did, it would completely destroy any possibility of having even a civil relationship and cause a firestorm of scandal. This is still Harry's dad and brother, and he clearly has a complex relationship with them.
First, if their Ducal titles were removed (which could be done pretty easily) they would automatically default to being Prince and Princess. I'm pretty sure the BRF and the media don't want a situation where Meghan's official title is Princess because that's a word with a lot of cultural weight, and would lead to increased comparisons with Kate now her official title is also Princess.
To strip them of all titles (ie to stop Harry from being Prince Harry) would require an Act of Parliament, and there's no way that wouldn't cause a huge worldwide scandal. Imaging being so vindictive to actually pass an Act of Parliament to remove your own son's birth title.
People would also ask why Harry was stripped when people who have done far worse were not stripped.
King Edward VIII abdicated and was a Nazi sympathiser and it's widely accepted that he was a Nazi collaborator who was actively a traitor to his country, he wasn't stripped of his titles. Prince Andrew was credibly accused of rape, continued a financial and social relationship with a convicted pedophile, had dodgy financial dealings, and was directly selling access to royals for money. He's still there and still a Prince.
The BRF's worldwide popularity right now is low, Will and Kate's last major foreign tour was rife with gaffe and accusations of racism, and countries around the world are either leaving or talking about leaving the Commonwealth. Outside the UK (especially in America) there's a lot of criticism a general perception that the BRF has a problem with racism. If you think "Being a Nazi collaborator or accused rapist with ties to paedophilia doesn't get you stripped of your titles, but marrying a black woman and leaving to ensure her safety does" wouldn't be a worldwide scandal, and wouldn't increase the perception of the BRF being racist, you're wrong.
Second, the royal family do not want them to lose their titles. They have an agenda and investment in them keeping their titles. Why? Because no child of a monarch has ever been stripped of a Prince/Princess title before in modern history, and doing so now would create a dangerous precedent that could end up being used to hurt Charlotte and Louis, or any of Will and Kate's future grandchildren. Will and Kate absolutely don't want to create a situation where if one of their kids goes through a sticky patch or struggles to find paid employment, or is perceived as a freeloader (and the media WILL be all guns blazing against the spares), the public's response is "well just strip them of their Prince/Princess title, you just did it to Harry."
If Harry and Meghan gave up their titles voluntarily it would be perceived as H&M snubbing and rejecting the RF's magnamity (which would be framed as, aren't the RF marvellous and gracious for letting those ingrates keep their titles after all the horrible things they've said, and those wicked people are throwing their generous offer back in their faces). They'd be accused of hypocrisy after what they said about not denying their children their birthright. There'd be no benefit.
Even if you strip the title of Prince from Harry, he'll always be the King's son, and the world will always call him Prince Harry the way the world always called Diana "Princess Diana" when she never held or was entitled to hold that title. So the BRF would be causing a lot of fuss and scandal for no reason.
If the BRF didn't want Harry and Meghan to continue to have titles, they wouldn't have them.