Right now, and for the next hundred years or so, Dogecoin operates like, and will continue to operate like, most other crypto assets. “Capped” assets are nowhere near mined out – and will continue adding to their supply, just like Dogecoin, for the foreseeable future.
Dogecoin’s supply is not unlimited, because it has an absolute limit of issuance per block, per day, per year – just like other coins do. The only difference is that Dogecoin’s issuance does not have an end date. Therefore, Dogecoin is only “infinite” over “infinite time”. Over finite time, its issuance is, in fact, finite.
This annual issuance is needed to pay miners and keep the network secure. Other chains, like Bitcoin, which theoretically will see their yearly issuance stop completely in 2140, will need to find a way to secure their network at that time (if it will indeed still exist), or the design of their consensus mechanism will need to change completely. Simply stated, putting a cap on Dogecoin would render the network insecure and vulnerable to attacks.