@sunflowersandbuttercups - that's my point. Friendships 'progress', not stagnate. You've spoken to your online friends on the phone, via FaceTime and Zoom. I also presume, irrespective of the gender of your online friends, you tell your partners/family/friends about them - whether that be mentioning them casually in conversation, showing a funny photo they've posted, or even saying "sorry, I can't cook dinner this evening, I'm planning on a zoom call with X". I also presume, to some extent, these friends are involved in your life and you in theirs.
The difference between your friendships and OP's friendship is:
a) As I said, friendships progress, not stagnate. This particular relationship went from being flirtatious, with long, deep phone conversations, to less frequent 'surface-level' messages and likes being sent largely by OP to the man. The OP doesn't know anything about this man 5 years on, other than what he chooses to publicly display on social media to her and every other follower he has. By definition, this therefore is not a 'friendship'.
b) The secrecy aspect: the wife clearly did not even know of her existence, let alone that they were still in touch 5 years on. They didn't (until yesterday) follow each other on social media. I'm sure, @sunflowersandbuttercups, that you speak to your partners, family, etc about your online friends, and that you update your online friends about those people who are important to you in real life when you have conversations on the phone with them. This aspect of that friendship is missing between OP and the man.
c) The one-sided dynamic: it seems like OP is the one initiating most of the communication here - "Occasionally he'll put a story or picture on his Instagram of his fiancé and kids I always like it" and "Other private messages included me congratulating him and her on birth of baby etc" vs "For the record he doesn't ever like any photos unless it's of my dog or me and my partner."
Even in this conversation:
"I put on my Instagram a recipe for my home made macaroni cheese and said guy messaged me and said "hi how are you? Stolen the receipt for Mac cheese looks amazing" I replied "I'm good thanks how are you? How is (sons and fiancées name) hope you're all well!" He replied "we are great thank you!" I double tapped it to like it."
He only got in touch with OP because of a recipe, and asked how she was doing out of politeness (notice that, unlike OP's replies to him, he doesn't ask about/mention her partner - presumably because he's not as invested in her life as she is in his). OP herself is the one carrying on the conversation, bringing his wife and baby into the conversation, and his response is to end the conversation pretty abruptly and not bother carrying it on. When he issues a final response that is impossible for her to respond to, she double-taps to give it a like instead, as evidence that she's read and acknowledged his message to her.
OP is not close enough to this man to have his number and message him directly (which would be an indicator that they are actually friends), nor to have ever got to know his wife directly (online) either.
Just because OP's partner knows about this, and just because OP herself is well aware that this man is engaged and has a baby, that doesn't mean that this is an appropriate relationship to have. As many people have said here, messaging somebody "congrats on your baby, hope your wife is well" on the surface of it doesn't look wrong. But it's the level of energy and time she's invested into staying in touch with an internet stranger, the fact that she's told him deep, personal secrets about her life, and years on carries on messaging him, the pleasure she gets from both giving somebody attention and receiving it in return when she gets a response.
As I said, the fact that she went to such pains to insist that this wasn't a flirty conversation implies that it could very much lead that way if both parties wanted it to; moreover, the fact that she's uncomfortable/posted on Mumsnet for advice because the wife found out they were in touch and followed her online also indicates a level of discomfort with somebody finding out about this relationship.
I hope this clarifies why I think it's an odd and weird 'platonic' relationship that I think should be stopped now. There is no reason for OP to still be so invested in an old online flame.