jq and array of two things: key and value

You can get pairs by using the to_entries filter. It will transform the dict into an array of items:

$ curl <url> | jq ".rates | to_entries"

[
    {
        "key": "btc",
        "value": {
            "name": "Bitcoin",
            "unit": "BTC",
            "value": 1,
            "type": "crypto"
        }
    },
    {
        "key": "eth",
        "value: {...}
    },
    },
    { ... }
]

Once you have such entries, you can access .key and .value.value of each item to get the pairs you want:

$ curl <url> | jq '.rates | to_entries | .[] | [.key, .value.value]'

["btc", 1]
["eth", 14.376]
[...]

The same for transforming them into a string:

$ curl <url> | jq -r '.rates | to_entries | .[] | "\(.key) \(.value.value)"'
btc 1
eth 14.368
...

Note that the select(.) you had before seems completely redundant...