Maps and vectors are very different.
The short version to the actual question you asked:
if all you do on your customized map is key-based lookup of already existing keys (the operator[]) and your push_back it may act like an inefficient drop-in replacement for vector where you only use the vector operator[] and push_back, yes.
The long version providing some background on why what you are doing is probably not actually what you want:
A map doesn't have an index, it has a key. A map is usually implemented as a red-black tree. Such a data structure allows for efficient lookup based on the key. You usually care about the key of a particular element, the key itself carries important information. Keys are usually not contiguous, and maps will not allocate space for keys that are not used in the map.
A vector is a contiguous block of memory. This allows for efficient indexed access. An index is not the same as a key: you generally don't care about which index a particular element gets, which index you do get depends on the order of insertion (they key is independent of the insertion order in in a map), indexes into vectors are always integer values, and you cannot have non-contiguous indexes.
If all you do in your map is your own custom push_back then to the outside it might appear to function like a vector in some contexts, and it might not in ways in other contexts (e.g. iterator invalidation).
Since you don't actually care about the key for the element that gets added in your example the choice of a map is pointless. Indexed lookup in the vector will be faster, and memory overhead will be smaller (though you could end up with memory fragmentation issues if you allocate very many objects, but that's a separate subject).
And finally, if you don't know which container class to use, vector and list are the places to start. Understand the differences between those two, and when you should use either of them, and then move on to the more advanced, specialized containers like map, set, their "multi" variants, and their "unordered" variants.