Doxycycline belongs to a class of drugs called Tetracyclines. Tetracyclines are unrelated to penicillins and therefore are safe to take in hypersensitive patients.
Other unrelated antibiotics include quinolones (e.g. ciprofloxacin), macrolides (e.g. clarithromycin), aminoglycosides (e.g. gentamicin) and glycopeptides (e.g. vancomycin).
Penicillin belongs to a class of drugs called beta-lactam antibiotics. These drugs include penicillin and amoxicillin. They are used to treat many common bacterial infections including skin, ear, sinus and upper respiratory infections.
Penicillin allergy is an overreaction by your immune system to penicillin and other antibiotics in the beta-lactam family and closely related antibiotics, including some cephalosporins.
For more information on penicillin allergy see:
https://www.drugs.com/mcd/penicillin-allergy