Are Mornington Peninsula towns considered suburbs of Melbourne and are they a part of the greater city of Melbourne?

There is a mental barrier for me, as a longtime resident of Melbourne: Melbourne ends at Mt Eliza. The mental barrier is to do with continuous buildings: there is a break in construction just south of Frankston, and there is empty land between Mt Eliza and Mornington (even if it is roughly 5 km).

The break in coverage of the metropolitan railway is another such construct: there is empty land between Dandenong and Pakenham (though not for long), but I think of Pakenham as part of Melbourne, because the train ends there.

There are people that commute into Melbourne CBD from the Mornington Peninsula, and for that matter from Geelong; but I would stick with the continuous building criterion. You are right that Melbourne has uncommon urban sprawl; and Sydney’s is worse. But I would think of Melbourne still as having a 50 km radius, not 100 km.