How do I upgrade my 2015 MacBook Pro hard drive?

You can replace the internal SSD in your 13" 21013 MacBook Pro with a larger one (see Other World Computing – OWC Solid State Drives For MacBook Pro with Retina Display (Late 2013 - Mid 2015)), but that will not allow your MacBook Pro to upgrade past Monterey.

There is a "gotcha" with Monterey and third-party internal SSDs. When you run Monterey, it reportedly checks to see if your firmware needs updating. If so, it runs a firmware updater. If that updater sees that the internal SSD is not an Apple SSD, it chokes, and both the firmware update and the Monterey installation fail.

Thus, you need to have installed Monterey somewhere before swapping in a third-party SSD. It appears that you have already done this, so I am mentioning this for the benefit of others who might be reading this.

There are some applications that you could use to extend the useful life of that machine.

These include

• Many third-party Web browsers, such as Firefox and Brave. The current version of Firefox will run on Catalina or higher. Several other major third-party browsers used to require Big Sur, but recently started requiring Monterey. There's a version of Firefox that runs on Sierra and that is still getting "critical security updates" for a little while – although it stopped getting new features a long time ago.
• The LibreOffice office suite
• The GNU Image Manipulation Program (GIMP)
• The commercial photo editing program GraphicConverter

These may help to substitute for applications that you can't get from vendors like Microsoft and Adobe who have a policy of supporting only the "most recent three."

Until recently, I would have also mentioned

• The commercial Affinity V2 applications (Photo, Designer, Publisher)

but the vendor has taken the unusual step of removing them from the market, in preparation for the launch of new versions or a new product.

just because a Mac is no longer supported, that does not mean that you must stop using it. when I got my current Mac, I gifted my old one to my dad. it still performs good, and he is very happy with it even though it cannot be upgraded past High Sierra.

just practice safe computing habits and you should be fine.

an excellent user tip to read is Effective defenses against malware and other threats - Apple Community.

Not past Monterey.

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No. Obtaining service for your Apple product after an expired warranty - Apple Support