Hi Mark,
Please check some of the answers below.

1. Is the database server the same? Or: is the OS difference only
client, only Firebird server or both client and Firebird server?
Linux 2.5.6.27020
Windows 2.5.3.26778

2. If not, is the DDL of both the same?
3. Are the Java version, Firebird version and Jaybird version the same?
Linux Java 8. Windows Java 1.7
4. How do you insert the value, specifically how is the timestamp
generated (created in Java, hardcoded expression like 'now' or
CURRENT_TIMESTAMP), default column, trigger)?
The value is inserted with JDBC PreparedStatement. Exactly the same code on Linux and Windows
5. How do you select the value?
I use the Timestamp column for ordering in the GUI and noticed that it does not work when fetching from Linux FB.
The select is made with JDBC into Resultset.
I made a SELECT with ISQL on Linux and confirmed 000 ms
6. What happens if you use CURRENT_TIMESTAMP or CURRENT_TIMESTAMP(3);
try both when inserting, and - separately - when selecting, that is, on
insert replace the value with hard code CURRENT_TIMESTAMP or
CURRENT_TIMESTAMP(3), or on select replace the column with the hardcoded
value (don't forget to add an alias with the original column name).
7. How do you verify the millisecond part is not stored; did you check
the millisecond part before storing?
Pls see  5.