Yes, if the DM allows it
The Linguist feat (PHB, p. 167) uses similar wording to languages learned through a character's background. For example, the sage background gets:
Languages: Two of your choice
In the languages section where the tables are detailed, the instructions are given (emphasis mine):
Your race indicates the languages your character can speak by default, and your background might give you access to one or more additional languages of your choice. Note these languages on your character sheet.
Choose your languages from the Standard Languages table, or choose one that is common in your campaign. With your DM's permission, you can instead choose a language from the Exotic Languages table or a secret language, such as thieves' cant or the tongue of druids.
While the first paragraph specifically refers to languages learned through your background and race, it does not say these are the only ways to learn languages, and since the wording is the same I would say that languages learned through other methods (including the optional feat rules) follow the same restrictions.
Note that languages allowed by the DM are not restricted to languages from the Exotic Languages table, but can include other secret languages, of which Giant Owl may be considered one.
As noted above, languages you can learn from the linguist feat must be one of these three things:
- Found on the Standard Languages table
- A language common in the campaign world
- Approved by the DM as a learnable language for you
Giant Owl is not a language on the Standard Languages table, and while I do not know your campaign world, I will assume it is not commonly spoken. In this case, Giant Owl is a learnable language only if the DM approves it as a language you would have had the opportunity to learn.