Yes, plants too close together do compete. They compete for sun, water and nutrients. Sometimes they will not even produce what you want. A good case in point is radishes. If they are too crowded, none will make the large root you are seeking. Any of the root crops need space if you want a large root. It is not hard to see that if you want a 5 inch onion, it cannot be closer than 5 inches to the next one. 6 inches would be better, however if you are after scallions, they can stand much closer.
Often there is good information on a seed packet about thinning and spacing for the variety. If it says thin to 12 inches, there is no use to put 50 seeds in that 10 inches. Put 2 or 3 just to guarantee germination of at least one. If the packet says it will plant 20 feet of row, it will actually plant about 100 feet if you plant correctly. The seed companies want you to plant more seed. They are in the business of selling seed.
Yes, you can and should thin your plantings if they are too dense.
Gardening at 5000 feet elevation, zone 4/5 Northern Utah, Frost free from May 25 to September 8 +/-