Here is another solution which uses any/2

map(select(any(.Names[]; contains("data"))|not)|.Id)[]

with the sample data and the -r option it produces:

cb94e7a42732b598ad18a8f27454a886c1aa8bbba6167646d8f064cd86191e2b
a4b7e6f5752d8dcb906a5901f7ab82e403b9dff4eaaeebea767a04bac4aada19

The SQL-style operators don't work well for me as a straight-forward selection mechanism; I believe they have a very specific use-case, for which they are uniquely suitable for, and for anything else they're (at best) clunky. At least that's been my experience. And I haven't really figured out what that specific use-case is, either.

With all of that as a backdrop, my recommendation is to use a simple regex test:

map(select(.foo | test("aaa|bbb")))

Given the example JSON:

<~> $ jq . /tmp/so4229.json
[
  {
    "foo": "aaa",
    "bar": 111
  },
  {
    "foo": "bbb",
    "bar": 111
  },
  {
    "foo": "ccc",
    "bar": 222
  },
  {
    "foo": "aaa",
    "bar": 333
  },
  {
    "foo": "ddd",
    "bar": 444
  }
]

the above filter would result in:

<~> $ jq 'map(select(.foo | test("aaa|bbb")))' /tmp/so4229.json
[
  {
    "foo": "aaa",
    "bar": 111
  },
  {
    "foo": "bbb",
    "bar": 111
  },
  {
    "foo": "aaa",
    "bar": 333
  }
]

If you need to generate the regex based on other data within the JSON, you can do that, too:

. as $data | map(select(.bar==111) | .foo) | join("|") as $regex | . = $data | map(select(.foo | test($regex)))

which would result in:

<~> $ jq '. as $data | map(select(.bar==111) | .foo) | join("|") as $regex | . = $data | map(select(.foo | test($regex)))' /tmp/so4229.json
[
  {
    "foo": "aaa",
    "bar": 111
  },
  {
    "foo": "bbb",
    "bar": 111
  },
  {
    "foo": "aaa",
    "bar": 333
  }
]

There may be a better way to run through the JSON twice (once to get the regex values, once to use it).