Yes: Use a user defined variable:
SET @position := 0; -- Define a variable
INSERT INTO products
SELECT id_product, id_category, name, (@position := @position + 1)
FROM db2.products
WHERE id_category = xxx;
The result of increment to @position is the value used for the insert.
Edit:
You can skip the declaration of the variable by handling the initial value in-line:
...
SELECT ..., (@position := ifnull(@position, 0) + 1)
...
This can be particularly handy when executing the query using a driver that does not allow multiple commands (separated by semicolons).