Gideon Bibles are not government handouts. They are absolutely 100% privately funded.
As nearly as I can determine after looking at a number of news sources from ABC to the AP to WAPO to several on line news sources, some Navy authority decided it was inappropriate to allow the Gideons to place the Bibles in Navy lodges and ordered them removed. Some sources referred to this as a 'removal of the Bibles' and some sources referred to it as a 'ban'. It seems pretty silly not to see the two terms as interchangeable.
However, apparently some Navy families noticed the absence of the Bibles and inquired and, when they found out the Bibles were prohibited from being placed in the rooms, they protested. The Navy reversed the order and allowed the Bibles to be returned to the rooms.
The Gideons have been around since 1899 and since that time have placed 1.9 billion and counting Bibles in hotel/motel rooms and have furnished Bibles to schools and colleges, prisons and jails, hospitals, and medical offices all over the world--they provide free Bibles to all who want or will accept them. They see their Christian ministry as sharing the word of God via speaking engagements, talking one on one with people, and making the scriptures available to all who need or want them.
If a Mosque or Islamic group or another religious group wanted to place their holy books in the lodges, then no doubt the Navy would also allow that too. To the best of my knowledge, nobody but the Gideons engages in that particular ministry.