Why does my 2013 iMac running Catalina say my software is up to date?

For one it would depend on which 2013 Mac you have. Big Sur requires one of these…

macOS Big Sur is compatible with these computers - Apple Support

Download and run Etrecheck. Be sure to give it Full Disk access before running.

Copy and paste the results into your reply. Etrecheck is a diagnostic tool that was developed by one of the most respected users here in the ASC and recommended by Apple Support  to provide a snapshot of the system and help identify the more obvious culprits that can adversely affect a Mac's performance.

Copy the report

and use the Additional Text button to paste the report in your reply.

Then we can evaluate the report to see if we can determine the cause of the problem.

This link will tell you the latest version of MacOS you particular Mac can run.

https://www.macworld.com/article/673697/what-version-of-macos-can-my-mac-run.html#:~:text=Catalina%20arrived%20in%20October%202019%2C%20and%20the%20latest,%28all%20models%29%207%20Mac%20Pro%20models%20from%202013

Apparently there is confusion between operating system updates and upgrades. An update is when we install a newer version of the same operating system. It will be found in the System Preferences by clicking the Software Update button. If you already have the latest version installed, you will be notified that no update is available That is likely what has occurred. An upgrade is when you install a newer operating system such as upgrading from Catalina to Big Sur. What you can install is dependent on the year and model Mac you have. Upgrade links may be found in the Apple App Store. I hope this information is helpful in clearing up any confusion.

Installing an update or an upgrade does not necessarily solve some problems such as running too slow. Most problems are caused by a specific issue and they are resolved by correcting that problem. Applying some of the following actions may be helpful in speeding up your computer. They are “maintenance” steps which are often ignored.

·      It is important to occasionally restart a Mac. Ideally, once a week or more often. This does some system cleaning.

·      Regularly clearing the history of your browsers can help as all the data is stored in RAM. Clearing frees up RAM for system use.

·      If your computer is set to startup with too many apps that will slow down the startup process.

·      Some programs use a lot of RAM. (Like many Adobe apps and the Chrome browser.) When using them, close any other apps not being used.

·      Doing a restart in the safe mode can clear a number of problems. This is done by holding down the Shift key at start up. This process can take up to ten minutes while the computer is doing its maintenance and cleans the cache. You may then restart normally.

·      It is also a good idea to run the Disk Utility app. If it reports some issues, then rerun it repeatedly as needed until the final report is ok.

·      On some computers the problem may simply be that there is not enough RAM installed for RAM hungry programs that are being used. The very minimum amount of RAM we would recommend for computers running the Catalina, Big Sur or Monterey OS is 16 GB. This information is found by going to the Apple logo in the upper left corner of the finder and clicking on About This Mac. Note; not all computers are able to have RAM user installed. Some upgrades must be done by a qualified Apple service specialist.

·      Last, and most important, you should remove any and all maintenance and antivirus programs. They will cause problems and will significantly slow your computer.

·      It is also very useful to download and install the free app EtreCheck. Be sure to enable full disk access and run it. Save the report. You can post the results here at the bottom under “Additional text”. It will tell us exactly what you have installed which might be problematic. Then we can offer some suggestions.

One final comment. If you have an older, slower mechanical hard drive installed it will be slow. That can't be changed. If that is the case, installing a faster external SSD and installing an OS on it, then making the SSD the startup disk can increase your computer's speed by 200 percent or more.

I hope these suggestions are helpful and enable you to speed up your computer.

To sort the fan issue try resetting the SMC two or three times in a row.

Reset the SMC of your Mac - Apple Support