If I try to refactor an object name with multiple words in it, only the one my marker is pointing at will be inserted into the textbox. But whatever I type in the textbox, whether it is multiple words or just one, is assigned correctly to the object.

I believe the new development environment needs the rules from the real world also in AL. No special characters in names. This means no spaces.
My recommendation camelCase.
We are aware of this issue but don't have a resolution yet. Currently this is a limitation in VS Code which only recognizes a single word as an identifier (which makes perfect sense for most other languages:)).
Note that it's only the proposed text in the UI that is incorrect. Whatever you type into that F2 dialog will be correctly used as a new name for replacing the full quoted identifier.
I wouldn't shed a tear when you decide to only allow single-word identifiers. And wouldn't that make things simpler for VS Code? For display reasons we have CaptionML property.
If there is one moment in time to make this change, then it's now! I must admit, there can be consequences we don't oversee. But still... the move to a new development platform would allow these kind of changes.
I am the opinion as Jan. It should be allowed in AL only one word Indetifier. No special characters.
Here are the VS Code issues for reference:
microsoft/vscode#37643
microsoft/vscode#7340
By the way, this was partially improved a few months ago. If you press F2 on the first word (e.g. Multi in the original example), then VSCode recognizes the whole identifier. For a complete solution (pressing F2 while standing on any word), we would need the above mentioned fixes from the VSCode team.
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I wouldn't shed a tear when you decide to only allow single-word identifiers. And wouldn't that make things simpler for VS Code? For display reasons we have CaptionML property.
If there is one moment in time to make this change, then it's now! I must admit, there can be consequences we don't oversee. But still... the move to a new development platform would allow these kind of changes.