Should I let the 2.4GHz and 5GHz bands on my router use the same SSID?

Normally both bands work with the same SSID and password.

Many dual-band routers can only use one SSID for both bands.

If you want to force one device to only use the 5GHz band then, if the router supports it, you need to use a separate SSID for each band.

that will only work if your devices ALL support dual channel connect (few do).

just add "5" to the SSID name.

Two different names. Makes knowing which one you're using much easier. You could make the names similar but have the endings something like <abc 2.4> or <abc 5.0>.

In your case, I'd say two separate names.

It's simpler to use a single name, but then individual devices could end up on either band if it supports both. Most will generally try 5 first then fall back to 2.4 but it's mostly out of your control. A device that only supports one or the other would end up on the one that it supports.

If you want to control which one ends up where, then you have to use two separate names. It sounds like your TV supports both but you want to make sure it's on 5 GHz, so you'd need to have two names and use the 5 network name.

Do you plan to use both?

If you only plan on using single (since it is replacing a single band one) one of these than sure. Just protect them by password or make a fixed ip connection between device(s) and your new AP.

Alternatively you can have virtual drives on that HDD and have each band use different part.

or you know just get another HDD i dont know what your use case is.