Can I use OpenStreetMap to make a map that contains lines and points (sort of like Google My Maps)

if I create a map using my own made-up data (such as a fictional road alignment or city limits)

You can use OpenGeoFiction for this, which is the OSM software repurposed for fictional mapping.

For A) the beginners guide or LearnOSM may be good starting points.

B) Fictitious entities are not permitted in OSM, everything should be verifiable.

First, is OSM only able to create maps by entering data, or can I manually place points and lines?

I'm not sure what you mean here, most software for editing OpenStreetMap is point and click.

Next, if I am able to place points and lines, if I create a map using my own made-up data (such as a fictional road alignment or city limits), is it forced to be in the real OSM map, or can I make my own map that's seperate from the one thats displayed when you actually use OSM to look at the map (if that makes any sense)?

OpenStreetMap's data is under a share alike license (ODbL), but OSM doesn't permit fictitious data so this could not be satisfied by re-uploading to the database in this case. I think you would have to keep it separate but I do not have the qualifications to be able to give legal or licensing advice. The FAQ may help you determine what is required.

THE OpenStreetMap is one big public database. Making any standard edits in one of the common OSM editors will result in them being reflected in that database. You should not add any private or fictitious data there.

Having that said there might be ways to achieve what you want. Unfortunately, it's not very clear what that is. A few options:

You can use uMap to overlay points of interest, ways, areas etc. on top of one of the many OSM based maps.

You could use the editor JOSM to make changes to the map but instead of uploading them to the public server you keep the edits locally and view them in JOSM only.

You could clone the OSM database or part of it (e.g. for a region) and make changes to that clone. Then set up some rendering software to generate maps.