I agree with you. The happy family illusion died when they divorced. Your household finances are none of his ex-wife's business.

Perhaps she has not entirely 'let go' and this is a plausible 'reason' to reach out. Agreed, your household finances regarding gift giving is not her concern. 

DH and I both have adult children. Both of our ex spouses have reached out and tried to stir the pot. We shut it down swiftly. Either could grasp the concept of moving on, especially DH's ex who was married and is now in the middle of a divorce, so both are blocked. Life is more peaceful.

It's due to a mindset of competition.  BM either wants to spend more or sabatoge DH somehow with the gift giving.   Do not engage.  Each house can do as they can afford.  The key is to make sure the skids don't turn into little narcissists over this.  

When DH divorced the ex wife. All of this goes away.  No discussion with the ex.  It's kids gifts ther her BF. And her car. 
I hope all of this is in text  not phone calls.   No phone calls all not deleted text for you to see ..

'Just another excuse for talking with the ex,  that's why no talking,  [_because there will always be excuses to talk. Kid in grown hair [ Texting only un deleted. [ you can read the 'nonsense between them] to see what texting about is really needed or just keeping in touch.  He divorced her and started a new relationship . Unfortunately is human sexuality relationship, once divorce, they can not go back to being just friends, especially after a new relationship.  I wanted nothing to with the ex. He was nice enough not to pay CS, so  talking. Joint anything was out of the question 

Do not discuss budget or presents with the ex. Do your own thing. Also do not tell her what gifts you'll be giving. (ask how I know... They steal the idea or try to upgrade it... Or duplicate it.) 

I think one of the basic red flags of a problematic ex is this kind of behavior--bad boundaries, an apparent hope for continuing control or relevance. It's unhealthy and unrealistic, and a very bad example to set for kids, who--let's face it--often hope that their parents will get back together. We dealt with a lot of this behavior early on. BM had a massive retaliatory fit when we put boundaries in place. DH can say, "You do you. We're all set. Thanks!" 

It may be something they did prior to you being involved.. perhaps in an effort to not duplicate or have one household spend in excess of the other.. in an attempt to make things seem fair.

She should not be setting your budget... obviously.. but if he wants to ensure they aren't buying the same stuff.. or if he is interested in keeping things somewhat equal.. if he wants to do it to.. then it may not be so much enmeshment but something they think is benefitting their kids..

if he doesnt want to engage in this exercise with her.. he is free to say.. Buy what you want and can afford.. we will do the same.

Xs have no say or standing in your marriage even if they are cobreeders with your SO.  Beyond the CO there is do duty of support, communicate, or collaborate.

BM can do what she wants regarding gifts when the SKids are with her including what they do for each other.

Ditto for your side.