Your husband is trying to bullshit you! Please don't fall for it. Don't accept his lies!
If a "random person" started messaging me these sorts of messages on FB, I wouldn't be replying.. I'd show them straight to my DP and say I don't know for sure who it is, but I have an idea that it could be xxxxxxxx.
Then, me and DP would decide together how best to deal with it. Whether that be by "playing along" to try and work out who it is, or by ignoring/blocking the account.
Either way, I would be very open about it to my DP from the start and if we decided I should "play along" to work out who the "random person" is, I would let my DP see the conversation.
Then... if the "random person" decided to send screenshots to my DP.. it won't be anything he's not already been shown by me.
Your husband DELETED the conversation.. that suggests he has something to hide!!! Normally, I would say most people do not routinely delete their conversations.
Where was your husbands message asking "Who is this?" There isn't one. Your husband knew exactly who he was talking to and they clearly have history. Blatantly an ex. He told her he still thinks about her. And when she suggests that things could have been so different (which is obviously referring to the fact that they could have ended up together).. your husband doesn't say "Yes.. but I'm now a happily married man!" He saying "....my fault", which implies some regret or some wrong doing on his part which caused the breakdown of their relationship.
This is not your husband "playing along" to try and work out who this person is. Your husband knew exactly who he was talking to!! This was a conversation reminiscing about a past relationship and what might have been... and he's deleted it to cover his tracks and now he'll stick to his bullshit story and lie through his teeth!
I'm really sorry OP, but your husband cannot be trusted.
Last year, an ex contacted me via FB messenger with an anonymous FB profile. He put up his mobile number and asked me out for drinks. I politely declined and he wrote back a flirty comment. I replied telling him I was in a relationship and it wouldn't be appropriate for me to keep chatting to him. He replied something snotty and I blocked him and never replied again. Didn't bother showing DP because it was so insignificant... but that convo is still there in my messenger history because A) I don't routinely delete convos. B) I'm not trying to hide anything because I've got nothing to hide. C) Little added bonus of having evidence to back up my honest and truthfulness in case the crazy ex ever tried to claim that convo was anything more than me rejecting him.
OP - tell your lying husband to get in the bin!