It is possible that a token you are interested in is not included in the token dropdown on https://uniswap.exchange/swap, however, all tokens that have a deployed uniswap exchange are supported on the front-end.
There are three ways to interact with tokens that are not yet included on the default list.
If a token is not included in the list, try pasting the token address into the search box. It will populate the dropdown with the token you are looking for.
For example, to populate the output token field with an unlisted token, we can specify the outputCurrency in the URL and pass in the token's address like this:
For tokens that have high market caps, high liquidity provision, or high dex trading volume, you can fill out a request to be added to the drop down menu on uniswap.exhange here: https://docs.google.com/forms/d/e/1FAIpQLSdQMI4KnQ1lCB0aiwzQ8xGTL59EX5FtkF6f2nT-JeQcxpW2Sw/formResponse
Token information (including name, decimals, symbol, etc) is pulled from token contracts directly. Logo images are pulled using the TrustWallet api. If you'd like your token logo updated make a pull request into the TrustWallet assets repo https://github.com/trustwallet/assets. Logos are found in ./blockchains/ethereum/assets/