MagSafe charger not working

There's a difference between wireless and MagSafe chargers, the MagSafe chargers can only be used with compatible iPhones (from iPhone 12 on) and compatible accessories, it snaps to the back of your iPhone or MagSafe case or any MagSafe accessories and it won't fall off, for this reason there's also a MagSafe wallet that sticks tightly to the back of the iPhones. The fact that you say your charger doesn't snap to the back of your iPhone it makes me think that it's not a MagSafe certified charger, or a MagSafe charger but it's more likely a wireless charger. So I suggest you buying a certified MagSafe charger, and try again. You can also to an Apple Store if it's close to you and try them on directly so you test your iPhone too

How to use your MagSafe Charger with iPhone 15, iPhone 14, iPhone 13 and iPhone 12 models – Apple Support

Note that a MagSafe charger for a watch is different than one for a phone and though they are theoretically interchangeable the magnetic ring for a watch is a different diameter than the one for a phone, and the smaller coil for a watch will require more precise placement of the phone on the charger.

You can see the different diameters on the photo of the now discontinued Apple MagSafe Duo Charger:

Is it an Apple original Magsafe charger, are you using it over a magsafe case?