You can copy places.sqlite (and possibly favicons.sqlite) on the old laptop to a USB (flash) drive and copy this file to the profile folder on the new laptop.

• https://support.mozilla.org/en-US/kb/recovering-important-data-from-an-old-profile

You can use the button on the "Help -> Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page.

• Help -> Troubleshooting Information -> Profile Folder/Directory:
  Windows: Show Folder; Linux: Open Directory; Mac: Show in Finder
• https://support.mozilla.org/en-US/kb/Profiles

You can copy files like these with Firefox closed to the current profile folder to recover specific data.

• bookmarks and history: places.sqlite
• bookmark backups: compressed .jsonlz4 JSON backups in the bookmarkbackups folder
• cookies.sqlite for the Cookies
• formhistory.sqlite for saved autocomplete Form Data
• logins.json (passwords) and key4.db (58+) or key3.db (57 and older) (decryption key) for Passwords saved in the Password Manager
  if you only have key3.db then make sure to remove an existing key4.db
• cert9.db (58+) or cert8.db (57 and older) for (intermediate) certificates stored in the Certificate Manager
  if you only have cert8.db then make sure to remove an existing cert9.db
• persdict.dat for words you added to the spelling checker dictionary
• permissions.sqlite for Permissions and possibly content-prefs.sqlite for other website specific data (Site Preferences)
• sessionstore.jsonlz4 for open tabs and pinned tabs (see also the sessionstore-backups folder)

Thanks, that worked!

But why doesn't Firefox sync it all automatic'ly? Why does it only bring over a small part?

It is possible that only the last few months (60 days) are synced and syncing is done in small chunks, so it can take some time to sync a large history.