Replacing a singleton solution for a memory manager in C++

I have in my project two classes for managing memory: MemoryStorage and MemoryFile.

The first works similar to a HDD and keeps MemoryFiles (files, directories, etc.) by using a singleton (add, remove, free, etc). The second works like fstream, but for memory (open, close, write, read, etc.).

Use Example

MemoryFile file(&MemoryStorage);
// Open file in write mode
if (file.open("filename", open_mode::out)) {
    // Data file
    int* obj = new int;
    *obj = 0;
    // Write in internal buff
    file.write(obj, sizeof(int));
    delete obj;
    // Put buff in MemoryStorage and clear buff
    file.close();
}
// Other moment
// Open file in read mode
if (file.open("filename", open_mode::in)) {
    // Data file
    int* obj = new int;
    // Write in internal buff
    file.read(obj, sizeof(int));
    cout << *obj << endl; // Print: 0
    delete obj;
    // Put buff in MemoryStorage and clear buff
    file.close();
}

If I don't use a singleton, creating files with the same name in memory will be possible, leading to an inconsistent file system.

I've read several articles in stack overflow like this one that talk about how singleton is bad pattern according to many programmers.

How else do I resolve the programming problem above without a singleton pattern? Do I use a private static member in MemoryStorage that replicates the effect of a singleton?