Laminate strips like that are often made of a material like the commercial product called Formica. The material was most likely originally installed with an adhesive called contact cement. This type of adhesive was painted onto both the strip and the edge of the counter top and let dry some before sticking the two parts together.
As you can see this type of adhesive can fail over time, especially when used on narrow areas like in your case. Unfortunately the presence of the old adhesive on both the strip and the counter top edge does severely limit the available choices of what you can use to re-adhere the strip. Epoxy would almost for sure not bond well with the old dried contact cement.
The best bet would be to use some stripper to remove the old contact cement before attempting a repair but that could be very difficult without removing the whole strip. It may be possible to apply an additional thin layer of new contact cement. It may soften the old adhesive enough that you can successfully rebond the end of that strip.