Paragraph Styles in Word document

I have a long Word document that I would like to continue working on in LibreOffice.
It has several user-created paragraph styles.
LibreOffice seems to correctly recognize the styles, but when I want to apply a style to a section of text, the style gets a different name.
Is there a way to keep the styles as well as their names, so I can apply them to new text without making a mess out of things?

Not enough information in your question to give a relevant answer. Remember that styles in M$ Word are much less powerful and elaborate than in LO Writer. Because of that Word tends to create many anonymous styles whenever a variation is applied to a paragraph. These styles received a name when working in Writer and the Word-induced inconsistency creates the mess. The key to success is correct identification of “real” original styles and apply only those, not the derived ones resulting from additions with ribbon buttons (or menu shortcuts equivalents). Your choice is between two methods: styles (then work only with styles without any temptation to “enrich” formatting with buttons like bold or italic) or manual formatting with toolbar/ribbon buttons. Needless to say that only the first one guarantees consistency.

I have never had such a problem. I import lots of DOC(X) documents and styles are retained with their respective names. Maybe, in your case it is about direct formatting messed up with styles. Or about pre-defined styles, their names may change depending on your locale. If you share a sample file, perhaps, the things will become clearer.

By the way, if you want to continue in LibreOffice, make sure to save your document to ODF.

Please explain how saving the doc to ODF affects paragraph styles.

Not much. But if you keep on working with a DOC/DOCX file, you will inevitably end up with a broken file. Work only in ODF.

When using styles in LibO writer you need to save and keep the file in international standard .odt format. Word does not support this function fully. Hence you will find a number of names used to store a single style used in various places in the document. If you have to use .doc or .docx to send it to a user with only Word and not LibO installed, keep it in .odt format and save a copy in .doc specifically to send only. We normally suggest you use .doc rather than .docx as Word seems to have fewer problems with that format.

To use my user-created styles I need to open “manage styles”, and that in turn opens a sidebar that I then need to close.
Is there a better way to do this?