Being married means doesn’t mean that you’ve now entered into a warden/jailer contract where all communications with all people can only occur with the approval and permission of your spouse. Because OP is bothered doesn’t mean he must fix it. People get bothered for unfounded, irrational, or “incorrect” reasons all the time and simply responding by giving in doesn’t solve the problem. In many cases, it makes the problem worse.
In a situation where one has an ex, you have to accept that for a period of time, your partner had a life that intertwined with somebody that wasn’t you. They shared friends, social circles, get to know family members, life events, and experiences... And that’s totally OK. That’s not something you can hold against your partner. Who says contact has to end? Plenty of people have relationships with people and realize that the relationship was a bust but a friendship is fine. In this case, it’s not even a friendship. It’s a polite nothing, an expression of a sentiment. He probably got the email just before a similar sentiment from Babies R Us and just after the invite from Ben and Jerry’s for a free, celebratory cone. It’s the absolute most passive, non-intimate way to express the feelings of the moment.
If an ex adds their name to the chorus of emails, letters, and personal congrats that come with major life events like marriage and baby... Well, good for them. Not only is it good etiquette, but it’s a testimony to the type of guy she married as most people don’t make it through a breakup where we’d be cordial enough to share such polite nothings. It means he’s a decent guy who still has the respect of somebody after what was probably a not-so-fun thing to go through. It’s quite a statement as to his conduct and his character.
As long as there isn’t some sort of other shoe here (he cheated with her, the congrats came with a hint at something inappropriate, or the response was hinting at something inappropriate), there is absolutely nothing wrong with the email “congrats” and the response “thanks.” OP said the full context of both emails was nothing but that. Celebrate the new family, a husband who is open and honest (another testimony to his character), and move along with life. The ex is an ex for a reason, and the wife is the mother of his child for a reason. The ex will fade back into the woodwork until the next life event while OP will keep building a life with her husband. Clear winner here is the wife.
I applaud the OP for considering that her response may be off to the situation and taking a pause. Seems like despite the hiccup, they have actually really open communication. Lots of people come on here and get upset that they make the demand their spouse doesn’t get because it’s a bit out there and their spouses respond by saying “yeah sure, won’t do it again” and then make a mental note that if the ex emails to not mention it (regardless of if they reply or not).