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Hi, I can help you with your question today.
Yes, a swelling under the eye is frequently attributed to an abscess. Especially in dachshunds which are known for bad dental disease!
You may be able to look at the largest back tooth on that side and see a fracture, pus at the gumline, and pain or redness near the tooth. It is not uncommon for an abscess to be undetectable without using x-ray though, so you may not see anything abnormal at all.
I recommend having him seen by your local vet. I would start with a physical and careful oral exam. Then, we can start a course of antibiotics (Clavamox or clindamycin are great for oral infections). I would expect to see a significant decrease in the swelling within the first 3-4 days. This can confirm our suspicion of an abscess, and then he can have a dental procedure to extract the offending tooth and clean the remaining teeth.