Your old media streaming device that you used with your old HD TV may or may not support 4K resolution. But it might not actually be something you need to upgrade anyway, as most 4K TVs these days are smart TVs that come with a variety of popular streaming apps.
TCL's 4K Roku TVs, for example, have a Roku streaming device built-in that supports 4K. In fact, the Roku-powered TCL TVs with 4K and HDR are actually better than Roku's own media streaming devices because the TVs support a better version of HDR called "Dolby Vision," while Roku streaming devices don't.
If you're wondering, the TCL TVs with 4K and HDR also support the basic HDR 10 standard.
Plus, if you already own an Xbox One S, Xbox One X, or PlayStation 4 Pro, you already have your favorite streaming apps built into each system.