I wouldn't be worried about someone having the service tag - it is not a threat.

To add to this I believe if you create an account with Dell you can bind devices (Via their service tag) to your account that way if someone did give them your service tag they would ask that they prove their identity (In theory) - this doesn't stop someone from making an account and doing it, it's just unlikely as it won't benefit them to do this. Regardless of that, the service tag is pretty public information for example if you're using a laptop all someone has to really do is get a glance at the bottom. I guess if someone really wanted to they could log a load of support calls for your machine but it wouldn't impact you in any way, Dell might just get a little annoyed. And if it resulted in them needing to send an engineer they'd ask for the address to go to fix the device. No, giving out your Dell service tag isn't a risk as such - all it can really be used for is logging support calls with Dell when you need help due to a fault with your machine.

Service tag to express code