Comparing MongoDB and Minio S3 is like comparing apple and oranges, but your question lead me to believe you are looking for a primary document storage. As others have pointed out, there is no convinient way of doing a search across multiple S3 stored documents. Thst is not the main use case of Minio S3. S3 is an object storage which is good for storing and retreiving large files. You can query data within a large json file, but if you want to change the data you need to upload a new file/object.
When it comes to cost, Minio S3 will be cheaper as it is basicly free. MongoDB is also free, but the capacity to search for data combined with ACID capabilities cost system resources like memory and data nodes, which can be expensive. A 9 node HA cluster from MongoDB inc will quickly cost you 300.000 euro in setup and licensing costs, but you can ofcourse set it up and support MongoDB yourself, which is free. Minio will not consume that much memory, but you will be maintaining the minimum of 6 data nodes which does have a cost. Both MongoDB and Minio comes with libraries which are easy to use and quick to develop against.