The other answers have confirmed that Postgres will do this with array fields, and it's good advice to follow. It's also in my view much easier to read than MongoDB's query language is!

  CREATE TABLE documents (name text, tags text[]);
  INSERT INTO documents VALUES ('Doc1', '{tag1, tag2}');
  INSERT INTO documents VALUES ('Doc2', '{tag2, tag3}');
  INSERT INTO documents VALUES ('Doc3', '{tag2, tag3, tag4}');

  SELECT * FROM documents WHERE tags @> '{tag1}';
   name |    tags
  ------+-------------
   Doc1 | {tag1,tag2}
  
  SELECT * FROM documents WHERE tags @> '{tag2}';
   name |    tags
  ------+-------------
   Doc1 | {tag1,tag2}
   Doc2 | {tag2,tag3}
   Doc3 | {tag2,tag3,tag4}

  SELECT * FROM documents WHERE tags @> '{tag2, tag3}';
   name |       tags
  ------+------------------
   Doc2 | {tag2,tag3}
   Doc3 | {tag2,tag3,tag4}

Postgres certainly isn't perfect, but it's usually a good answer to "how do I store and query data" where you don't have any particular specialist requirements.