As @Fumblefingers notes, "abandon ship" is akin to a "phrasal verb". While it seems to be a slightly different form of fowl, it's apparently non-standard construction 'makes sense' in the same way that phrasal verbs do.
Importantly "abandon ship" and "abandon the ship", in the manner that they are understood and used in practice, are not directly equivalent, although they are usually (but not always) closely inter-related.
With "Abandon the ship" the focus is on the ship. The people doing the abandoning will often be important in the overall order of things, but to a variable extent. When the decision was made to abandon Shackleton's ship "The Endurance" because it was irrevocably trapped in pack ice and was being slowly destroyed by ice pressure, the decision was to "abandon the ship". despite this being during a period (mid WW1) when the term "abandon ship" was in widespread use, they did NOT "abandon ship" then or at any other time in the sense almost always intended by the term. They were already living on the ice, had removed many of the stores and had already realised that the ship was probably beyond saving.
- On October 27th Shackleton wrote, "The position was lat. 69°5'S, long. 51°30'W. The temperature was -8.5°F, a gentle southerly breeze was blowing and the sun shone in a clear sky. After long months of ceaseless anxiety and strain, after times when hope beat high and times when the outlook was black indeed, we have been compelled to abandon the ship, which is crushed beyond all hope of ever being righted, we are alive and well, and we have stores and equipment for the task that lies before us. The task is to reach land with all the members of the Expedition. It is hard to write what I feel".
"Abandon ship!" and "to abandon ship" are focused entirely on the people involved. It (usually) signifies urgent emergency action, often with danger and possible loss of life. The action may have consequences for the ship, and are usually necessitated by events relating to the ship - but "it's not about the ship".
In this Google Ngram chart
for the terms: abandon ship, abandon the ship, jump ship, heave to, heave ho
SHIFT-click on image for larger version
we can see a sudden peak in the term "abandon ship" in the 1914-1918 period (WW1), falling off over the next 5 years or so, then peaking again during the WW2 period, and then continuing at a much higher than WW1 level with several peaks (reason unknown to me).
Conversely "abandon the ship" had a WW1 period peak but has been fading ever since - presumably as the "abandon ship" = people-focus concept gained strength.