Do Prompto's photos have a practical/mechanical purpose (in and of themselves)?

TL;DR:

Yes and No.

It largely depends on your purpose for playing and how technical and meta you wish to view the intention.

If you're playing to say you played it and just want to push through in a rote sense, or hit a point where you aren't compelled to 'feel' about the game and it's characters:

No there is not practical gameplay value to them.

If you look for immersion and prefer to allow the game to draw emotional response in a way the developer intends and hopes, then:

Yes (arguably).

Explanation:

Discard without review doesn't directly impact anything in terms of trophy / achievement but reviewing photos is a necessary step in how they're saved (review is triggered by selection of the photo, which is why you'll hear banter over some photos and trigger the acquisition of an trophy / achievement).

The option to save photos indirectly impacts other portions of the game, specifically the ending sequences in ways that can be major or minor depending on how you value things like storytelling and emotional investment (something the game pines for from the audience).

Prompto's photos do have mechanical purpose in a meta sense. The series of side-quests that require Prompto to take photos and yields very high Gil reward, which is technically a mechanic of the game using the photos and the player is forced to review (but not save) them upon completion of each portion of the side-quest.

Additionally, certain locations and landmarks give the option to take photos, some Prompto will notify the player about, which builds a sense of how the characters interact with one another without really impacting their response in a sense of the mechanics you are most likely thinking about. Prompto will also occasionally ask Noctis who he prefers pictures of, and that directly impacts a meta mechanic, and arguably practical in nature.

A direct mechanic impact as well as practical impact of photos is revealed prior to the final boss fight.

Noct will have an opportunity to choose his favorite photo as a keepsake before heading in to fulfill his destiny. This, and all other saved photos will adorn the ending credits as a slideshow of what the player accomplished with the keepsake photo becoming the certificate of completion image for FFXV.

Thus, why not reviewing and deleting photos for the first play-through can lessen the experience of the game for some players.

NG+ Depth:

When / If you start NG+, all saved photos and the certificate of completion will remain in the Archives menu under Photos. Deleting photos altogether or deleting and replacing them upon the next round of completion will just alter what photos are used in the ending credits.

NG+ has a purpose many seem to immediately overlook and it's in the power boost gives the game a pace in-line with theatrical flow and presentation more than when you start out to have the FFXV RPG experience. This is especially true with the movie King's Glaive as a preamble to the game.

Given the ending credits entail the saved photos, reviewing them allows you to choose how you remember the journey. It's left to you to give it practicality from the established layered mechanic, more so if you felt underwhelmed in the first play-through with knowledge of what's coming.

No

The OP has already pointed out the few minor bonuses that taking pictures can provide the player. Shawn's answer does point out an end game action using the pictures, but that has no practical game play purpose.