I am in the UK and my daughters both paid for their mobile phone bills right from first getting them when they were teens. But I have often observed that in the UK people seem to favour this more, while in the USA (where the majority of members here are from) people expect to foot the bill for phones and cars etc for their children.  If your SD is 18 and has a job I would consider it only fair that she pays her own phone bill even if she is not paying towards household expenses. 

This is a parenting decision that has no right or wrong, lots pay for kids to have cell phones until they are in full-time adult jobs (especially since it's usually a family plan so all one bill), and others have kids pay younger. 
 

It's part your money, though,  so you should get a say in this, whereas normally I'd suggest parenting decisions be left up to the actual parent. This is why I think joint accounts when stepkids are involved aren't a good idea. 
 

Push your wife to compromise, or perhaps separate finances if you can't come to an agreement about how much to support adult kids financially. My guess is that this is just the start of the disagreements on this matter.

Lots of families may cover phone and other expenses like car insurance, health insurance etc.. on their adult children when they are just starting out and/or studying at a university or trade school full time.

So, what your wife wants to do for her child isn't wrong.. but she should be financially (your wife) a larger share of that bill.  In fact always should have been doing that.  

Don't get me wrong, there is also nothing wrong with parents that try to teach kids some responsibility by making those kinds of costs the child's obligation.  But if it is not financially hurting her parent, and if the girl is generally a responsible person.. is working and pursuing a degree with honesty.. then I don't see it as detrimental to her to help her out with that bill.

You and your wife need to devise a plan now for the expenses of your SD. It could be your wife pays from her funds if you don't agree with funding the phone. 
 
We paid for the kids' phones through college as long as their grades were good. Both had jobs but as their mom (dad and stepdad), we wanted to help them excel. Both had large scholarships with little cost to us for their education so that was another reason.  As you said your SD is saving for University, I see nothing wrong in paying her phone bill to help her out. 

Separating finances seems like a good solution otherwise you will be on the hook for many more expenses down the line. I think it just depends on the family. My husband's mom paid for his car and health insurance until he was 24. He paid for his phone but his sister did not pay for her phone until her early 30's. My family was a lot different as soon as I got a job at 16 I was pretty much on my own except they still let me live there. I bought my first car, paid my own insurance, bought my own groceries, and had a cheap phone provider to get me by. It was tough but it taught me to be responsible with my money and job.

Is the cell phone bill the real issue here? Or is it a flashpoint for you?

A parent's job is to teach responsibility and independence to their child. Is that happening? Is the daughter expected to pay for other expenses, such as her own clothes, gas for her car, entertainment, etc.? Or is Mom still handing out money for that?

You might want to look at the bigger picture here -- is the girl learning financial responsibility? How does she spend the money she earns? How much support do you and your wife provide, and how does your wife see that tapering off over time?

I don't think I'd make the cell phone bill my hill to die on, but I would insist that there are expectations of financial independence and responsibility, and some movement toward that goal.

My husband has been paying for SD's cell phone for years (to the tune of $100 a month!).   This made sense when she was in college. She is now in her mid=30's, married with kids and he's still paying with no end in sight!  They are not struggling financially, in fact they have nicer house,  newer cars,  go on more vacations, etc.  And my husband wonders why I don't want combined finances!