If someone told you you are a ‘hiring manager’ but you don’t have responsibility for the budget that hire comes out of, they are lying to you to make you feel important.
Hiring requires money. If you have to go ask someone else for that money, they are the one doing the hiring.
I can tell you first-hand, that's common. I control the hiring but not the ultimate say in the budget. Up to $X I can approve the hire, up to $YY I need to seek VP approval, up to $ZZZ I need SVP approval, up to $LOTS I need CEO approval. But all of those are feasible, given the right candidate. There is no hard ceiling on the budget, it's all about who I can convince.
The just a wee bit pedantic. In my position, I control my employees' salaries (within guidelines). But, if I want to add headcount I need to make my case to my VP or higher. I suppose I could fire 5 seniors developers and hire 10 junior developers with the freed up budget, but I definitely cannot conjure up the $200k+ required to hire an additional employee without approval from my boss or his boss.
Hiring requires money. If you have to go ask someone else for that money, they are the one doing the hiring.