I don’t expect either of them did. Teams tend to use the names of local professions, animals, or other recognizable local features. As long as the name is unique in the same league… “oiler” is an English word and not the property of anyone.

Searching your question, maybe they both were inspired by the naming brilliance of the Tulsa Oilers, as I see that “the Tulsa Oilers have existed as four separate franchises in five different leagues over their 90-year history.”

The Houston Oilers played in the AFL (before the AFL and NFL merged) from 1960 until they moved to Tennessee in 1996. The Edmonton Oilers were founded in 1971, eleven years after the football team.

The franchise changed their name to the Titans when they moved to Tennessee.

Some background:

Edmonton Oilers
When Bill Hunter was deciding on a name for his WHA team in 1972, he first called it the Alberta Oilers. Oilers was a nickname for his Junior A team in Edmonton, the Oil Kings; in 1973, Hunter changed the team’s name to the Edmonton Oilers. When the NHL absorbed four teams from the WHA in 1979, the Oilers retained their name.
Oil is one of Alberta’s most prevalent natural resources; the province has the world’s third-largest reserves behind Venezuela and Saudi Arabia, according to its website.

Houston Oilers
The Houston Oilers began in 1960 as a charter member of the American Football League. They were owned by Bud Adams, a Houston oilman, who had made several previous unsuccessful bids for an NFL expansion team in Houston.