It depends what limits your airline puts on hand luggage. If it's one piece only then no, if they allow 2 pieces then size permitting it may be ok. depends on the size. We do take a cool bag we bought at Aldi (pic below) counted as a personnal item, but it depends on the airline and your allowance.

You can take a cool bag as hand luggage provided it meets the standard size and weight restrictions of your airline. In 2026, most airlines allow a personal item (approx. 40 x 30 x 20 cm) or a standard carry-on (55 x 40 x 23 cm); if your cool bag fits these dimensions, it is treated as a regular piece of luggage. However, the contents are subject to strict "liquid and gel" rules. If you use ice packs or gel packs to keep items cold, they must be completely frozen solid when you pass through security; if they are slushy or melting, they will be confiscated as "unauthorized liquids." Additionally, any food items inside that are liquid or gel-like (such as yogurt, hummus, or soup) must be in containers of 100ml or less. If the cool bag is for medical supplies or baby milk, the liquid restrictions are waived, but you should declare the bag for manual inspection at the checkpoint.