Indeed. Egyptian archaeologists find him mentioned long before Ptolemy in artifacts from Rhacotis and Memphis and construed his name as a compound of Osirus and that of his earthly incarnation, the bull Apis. Other traditions connect Serapis with Seleucia in Cappadocia or with Babylon. It may be that a false etymology, confounding a hill near Memphis with the name of Sinope, was the source of the tale in Tacitus. See Samuel Dill, "Isis and Serapis in Roman Society" in Roman Society from Nero to Marcus Aurelius, pp. 561-2. Basically, the origin of Serapis is still a mystery and I have to acquiesce to simply acknowledging that it remains uncertain how his mythology came about.
What features determine a figure must be Serapis and without which a figure can not be him? Are there no local or other deities that can be confused with him? I see a trend toward calling any figure with a high hat by his name but do not know what evidence supports this. This was Pluto when purchased. and Hades?
What do you think: Is Serapis laureate too or is the tie seen on his head only used to fix the kalathos?
Well, any coin, really. Cerberus is associated with the underworld in mythology and it's only natural that the associated god would be Pluto.
Serapis is, among other roles, a god of the underworld and as his syncretic depictions progressed, isn't it natural for him to be depicted with Cerberus?
It could very well be, but it doesn't have to be. That's Doug's point, I think, that we don't know for sure sometimes which deity was originally intended.
The famous statue of Serapis from Bryaxis in the Alexandreian Serapeion already has shown Serapis with Kerberos as Roman copies demonstrate (Here a Roman copy from a museum in Liverpool). Therefore for completeness we have to say that Ptolemy I has made Serapis of Osiris, Zeus and Hades/Pluto. To the coin of Caracalla I have to add: Caracalla has erected a Serapeion on the Quirinal in Rome. Temples for Hades/Pluto admittedly are known for Greece but not for Rome.
@Roman Collector Thank you for the information. According to Michaelis's classification, the three Serapis standing coins in my collection that I'm sharing belong to Type 2. Although from your post I don't see that Michaelis has made a distinction between transverse scepter and scepter held away from the body. Also, we have the how but not the why: why is Serapis represented in these different ways? Without the "why" the classifications are interesting but maybe they're just stylistic choices of individual engravers without any larger significance. Caracalla. AR denarius, Rome, 213 CE; 3.29g. BMCRE 50, Hill 1359 (C), RIC 208a, RSC 211. Obv: ANTONINVS PIVS AVG BRIT; head laureate r. Rx: P M TR P XVI – COS IIII P P; Serapis, wearing polos (RSC says modius), standing front, head l., raising r. hand and holding transverse scepter. ----------------------------------- Caracalla. AR denarius, Rome, 216 CE; 3.19g. BMCRE 168, Hill 1532 (C), RIC 280c, RSC 348. Obv: ANTONINVS PIVS AVG GERM; laureate head r., Rx: P M TR P XVIIII COS IIII P P; Serapis, wearing polos (RSC says modius), standing front, head l., raising r. hand and holding scepter. From the Dieter Braun Collection. ----------------------------------- Caracalla. AR denarius, Rome, 216 CE; 19mm, 3.69g, 12h. BMCRE 168, Hill 1532 (C), RIC 280c, RSC 348. Obv: ANTONINVS PIVS AVG GERM; laureate head r., Rx: P M TR P XVIIII COS IIII P P; Serapis, wearing polos (RSC says modius), standing front, head l., raising r. hand and holding scepter. ----------------------------------- Caracalla. AR denarius, Rome, 217 CE; 18mm, 2.89g, 7h. BMCRE 176, Hill 1568 (R4), RIC 291b (C), RSC 385. Obv: ANTONINVS PIVS AVG GERM; head laureate r. Rx: P M TR P XX — COS IIII P P; Serapis, wearing polos on head, seated l., holding uncertain object (corn ears?) and scepter. Very rare; 1 in Reka Devnia. From the Laurent Bricault Collection of Isis and Serapis in Ancient Coinage. ----------------------------------- Caracalla. AR denarius, Rome, 215 CE; 21mm, 3.16g. BMCRE 125, Hill 1491 (R3), RIC 261a (S), RSC 299. Obv: ANTONINVS PIVS AVG GERM; laureate head r. Rx: P M TR P XVIII COS IIII P P; Pluto/Serapis, wearing polos (RSC says modius), seated l., on stool, holding scepter, and extending r. hand toward Cerberus at his feet. Rare; 3 in Reka Devnia. This type for Caracalla was present on antoninianii in both the Arnold (lot 119) and Kelly collections (lot 1251, first of two coins). However, both collections lacked examples of the Pluto type on denarii. Reverse die link to Naumann 58, 1 October 2017, lot 482 (realized hammer €260).