There are basically two ways to purchase Doge.
You can sign up for a coin based wallet. These charge a small transaction fee every time you buy, sell, or make a crypto transaction. In this case, you own the Doge just like your money in your bank. Unlike your bank, the wallets charge you the equivalent of a debit card foreign transaction fee for every transaction.
Second main method is what many brokers now offer, including Webull and Robinhood. They run the equivalent of an ETF. Just like the ishares SLV ETF to invest in silver value.
These brokers do not charge a fee, in exchange they cover your investment in their bucket of holdings. They take all the risk of being hacked, losing keys, etc.. I believe there are micro fees we do not see, via the actual wallet they are partnered with. Thousandths of a share for example, versus wallets that hit you for tenths on both sides of owning and selling/transferring.
By the way, almost every wallet allows other holders to short sell against the pool of Doge held. Yes, and they charge the short sellers hourly margin interest plus fees. Amazing