Is it possible to automatically generate a Unix millisecond timestamp in MySQL 5.7
MySQL 8.0 supports the expression default like you show.
MySQL 5.7 does not support expressions as default. Only NULL, or a constant value, or CURRENT_TIMESTAMP if it's a DATETIME or TIMESTAMP column.
For MySQL 5.7, you have two alternatives:

Declare the column as created_time datetime(3) default current_timestamp(3).

Use BIGINT as you are doing, but write a trigger to set the value.