Did the German Enigma machine use multiple encryption keys?

There were multiple Cypher Keys for example Supply Ships sheltering in neutral ports used the Tibet Cypher. merchant Raiders used Special Cypher 100. Cypher Bertok was used to communicate with the Naval attache Tokyo. Cypher Freya was used between OKM and naval shore installations. Triton Cypher was used by U-boats in the Indian Ocean.

There were even special Enigma keys created for co-operation between the German and Japanese navies operated on three rotor machines known as T-Enigma under the Tirpitz agreement of 1942 (Tirupitsu to Japanese/JN-18 to USN) The code key for this was originally known as Sumatra and later as Gartenzaun. This was mostly used by German I-class U-boats which sailed to Europe such as I-29 and I-52 for example.

The SS had their own Cypher keys such as Kliest many of which were rarely broken. German Railways had their own Cypher Keys as did various Luftwaffe units. The Kreigsmarine used four rotor Enigma machines as opposed to the three rotor machines of the Luftwaffe.

These were books of Cypher keys relevant for specific days and the way they were used changed as the war progressed. Generally a vessel was only issued with a book of Cypher keys lasting three months. Later in the war Keys would change up to twice per day. They were issued on pink water soluble paper.

The first Enigma machine to come to the notice of Germany's foes was in Poland, around 1928. Polish customs (and their intelligence service) were suspicious of the German embassy's unseemly desire to get a certain package out of Customs on a Saturday. The Poles spent the weekend copying the manuals and examining the mechanism, and then delivered it in the Monday morning post. In August 1939, facing invasion, the Poles gave all their knowledge (including 10 years of mathematical advances in code breaking) to the French and British.

You should watch the BBC series "The Secret War" (circa 1975), especially the episode "Still Secret", which covers Enigma/Ultra/Fish and Bletchley Park very thoroughly. The Allies were NOT dependent on capturing Enigma machines -- they made tremendous advancements in code breaking without having the physical machine or both plain and encrypted messages.

Yes, different keys were used on several levels. In an Enigma system, the key consists of several components:

1. Rotors (which rotor is used in which position). Originally, 3 rotors were used (each with unique wiring). Later on, more rotors were added (up to 8 by the end of the war).
2. Ringstellung (the position of the ring on each rotor)
3. Rotor starting point
4. Plugboard settings

In addition, not everyone used the same key:

1. Some groups (the Navy and Abwehr, for example) used their own rotors with different wiring.
2. Each group of operators (one army, or one U-boot group, for example) got its own list of keys.
3. Keys were changed daily.

Yes, they do allow multiple keys. Typically they had a new key every day.

See this Wikipedia article:

Though Enigma had some cryptographic weaknesses, in practice it was German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed.

It was not capturing the hardware alone that enabled the decryption. The key tables were books of keys to be used on a given day.

"Allies" capturing Engima machines (what you really mean was British navy, who then in Hollywood were magically transformed into US navy) was really of no importance. What was important was capturing code books. The wiring of the Enigma machine was known since the 1930s, when Polish mathematicians managed to reconstruct it from very limited information.

The latest Enigma machines came with a set of eight wheels, of which three were plugged into the machine, plus a fourth extra wheel for submarines. There was a "day code" which told the personell which three wheels to pick, how to rotate them, and which nine cables to use on the plugboard. When a message was sent, sender and receiver put their enigmas into the day position. Then the sender would send a random three digit key, and then both sender and receiver would switch the wheels to that key.