Is the wall flat? If an uneven wall surface is causing the lippage, replacing the tile won't correct the problem.

Yes it is flat. There was no backsplash there previously.

Where did the tiles come from?

Tiles are from Lowe's https://m.lowes.com/pd/allen-roth-Venatino-polished-Brick-Marble-Mosaic-Subway-Wall-Tile-Common-12-in-x-12-in-Actual-12-in-x-10-in/3502404

To me, it looks like it would need to be removed and re-done to really get it correct. Sometimes lighter grout (white) would hide imperfections more than a darker/ contrasting color, but it would not really be worth the energy to re-grout it (In my opinion). Although, it's an option to try re-grouting lighter, but don't expect too much. If the tile came out of the box with edges chipped, or surface warped, you may have a warranty claim. You would have to contact the manufacturer if that is the case. Did you DIY this, or hire someone?

Unfortunately we paid our usual handyman and assumed he could do it because all other work has been done well. While there are a few noticeable chips, there are so many edges that appear chipped but really I'm pretty sure it is grout left on tile. I checked all boxes for chips so I'm also wondering if maybe he used a metal tool to grout? Obviously I really regret not watching along the way and noticing it was crooked before he grouted. I did see it but I didn't notice the variation in depth against the wall or in spacing.

How much would a project like this typically run? We paid $325 for tiles and materials and $420 for install. It is 25 sq ft. Curious so if we redo I know what to expect for a proper tile install.

Also, could we grout over the existing grout adding another layer? Since it is very shallow grout, not sure if that could help?

Sarah- I was going to suggest discussing the issue with your tile installer, and explaining that you will not accept the product. Some people use payment as recourse.

Normally, the grouting is done with a rubber or plastic float, then sponged down. It would be unusual for any metal tools to be used. It really looks like it was cut, perhaps to try and correct something for straightness? It's plausible a metal tool was used to try to scrape down uneven grout after it was dry.

In any case, if you leave it as is, you may want to place a bead of clear silicone at the countertop so it is a bit more watertight.