How about using a regular tag with a number and keeping all the info. on your phone, computer or even in a paper file. Giant plastic tags in the garden sound really ugly.

I stopped to view the landscaping in a large corner lot who used plant tags about the size you're looking for and they didn't look ugly. He had 150' on both sides and mimicked a natural forest ecosystem and the best I've ever seen in person or in pictures. I believe they added the big markers with the common and scientific names of plants for people so they wouldn't come to their door asking the name of a plant. It was an attraction and I returned many times.

What you're looking for more then likely will need to be custom made for you or made by yourself cut out of UV stable and durable plastic sheet like the product in the link.

https://www.menards.com/main/building-materials/panel-products/plastic-utility-panels/1-4-x-4-x-8-hdpe-panel/whtkhdpe48x96x220/p-1561703293086-c-14048.htm

What may also work (and cheaper) is this 5/16" thick flat PVC trim which is smooth on on side if 5 1/2 inches is wide enough for your needs.

https://www.menards.com/main/hardware/garage-door-parts-accessories/5-16-x-5-13-16-x-8-pvc-flat-utility-mouldings/4179869/p-1642874329074627-c-7577.htm?exp=false

PVC trim is what I cut up to make 10" long and 1"+ wide plant markers because no one sold them and I didn't want wood or metal. Works good and 99% rubbing alcohol will take off the permanent black marker when I want to change them to a different plant name. I grow multiple varieties of the same vegetable and often got them mixed up, or couldn't find or see the little cheap markers sold under the foliage. Anyway it's what I wanted.

@outpaientzero I am going to try these tags:

https://www.amazon.com/dp/B0DRV979CH