There isn't a specific field for secondary addresses currently, but you can add the secondary address to either a pop-up or custom memo as these are reportable in a Custom Entities Person's report which is likely the main report you would use to pull information relating to people records.
The only real difference between using a pop-up memo and a custom memo type you configured manually is that the pop-up memo will display on the animal's record immediately upon opening it as well as on the memos tab, and all other memos will only exist within the memo tab of the record. Memos of both types are entered in the same way, and memos are very quick to add so it shouldn't be too much trouble to navigate to the memos tab to add the additional address details when needed!
If you do need to know immediately upon opening a record if there is a secondary address stored, I recommend using a pop-up memo since they are not easily missed. You could also create an attribute called Additional Address (or something similar) that you can add to the person record as a "flag" to indicate that there is another address stored in a memo.
The other thing to keep in mind here is that when you include memos in a custom report, the report will only pull in the most recent memo that was added for each memo type you include in your report. If you have multiple pop-up memos stored on a record but the one relating to the address isn't the most recent, you won't be able to pull the address memo into reports.
For this reason, you may want to consider creating 2 different custom memo types - one for Mailing Address and one for Physical Address. You can add a P.O. Box address to a person record as the primary address if you'd like, and in this case, I would recommend adding the physical address to a Physical Address memo that you can pull into reports.
It's much less likely that you would have multiple memos of either type since they will likely only be used once to store one additional address, and having 2 options to differentiate between mailing address and physical address might be helpful as well.