Notice the constraint in the docs when using --db-instance-indentifier:

--db-instance-identifier (string)

  The user-supplied instance identifier. If this parameter is specified, 
  information from only the specific DB instance is returned. This parameter 
  isn't case-sensitive.

Constraints:   
  - If supplied, must match the identifier of an existing DBInstance

So you can only use this option if you know the DB in fact exists.

Using queries

To search for a DB that may or may not exist you'll have to use the --query option:

$ aws rds describe-db-instances \
    --query 'DBInstances[*].[DBName,DBInstanceIdentifier]' --output text

The DBINstances JSON structure is accessible in the awscli help:

$ aws rds describe-db-instances help --output text
...
...
       {
          "DBInstances": [
              {
                  "PubliclyAccessible": false,
                  "MasterUsername": "mymasteruser",
                  "MonitoringInterval": 0,
                  "LicenseModel": "general-public-license",
                  ...
                  ...
                  "DBName": "sample",
                  ...
                  ...
                  "DBInstanceStatus": "stopped",
                  "EngineVersion": "5.6.27",
                  "AvailabilityZone": "us-east-1e",
                  "StorageType": "standard",
                  "StorageEncrypted": false,
                  "DBInstanceClass": "db.t2.micro",
                  "DbInstancePort": 0,
                  "DBInstanceIdentifier": "mydbinstance-1"
              }
          ]
      }
...
...

Using Filters

Another simple solution for the initial question is, to use the --filters parameter. The query will return either the instance identifier (if the instance exists), or an empty string (if it does not exist):

#!/usr/bin/env bash

DBINSTANCEIDENTIFIER="greatdb"
EXISTINGINSTANCE=$(aws rds describe-db-instances \
    --query 'DBInstances[*].[DBInstanceIdentifier]' \
    --filters Name=db-instance-id,Values=$DBINSTANCEIDENTIFIER \
    --output text \
    )

if [ -z $EXISTINGINSTANCE ]
then
    echo "instance $DBINSTANCEIDENTIFIER does not exist!"
else
    echo "instance $DBINSTANCEIDENTIFIER exists!"
fi

References

• https://docs.aws.amazon.com/cli/latest/reference/rds/describe-db-instances.html
• https://stackoverflow.com/questions/45449174/how-do-i-use-query-parameter-in-aws-rds-describe-db-instances-command
• https://stackoverflow.com/questions/46051538/syntax-for-filters-for-aws-rds-describe-db-instances

You can do something like this:

aws rds describe-db-instances --query 'DBInstances[].DBInstanceStatus[]'

output:

[
   "available"
]