This question has broad applicability to all outdoor use of a Pi, so I will cover it quite expansively...
Although possible to use a USB powerbank, as @goldilocks notes, batteries will have to be changed frequently and if the goal is to monitor nature without interfering with it, going to the birdbox frequently to change batteries will disturb the birds, potentially keeping them away.
POE is really easy to implement, and once your system is setup will have zero human overhead required to maintain it, running unattended 24/7/365. Even while you're away on vacation which wouldn't be possible with batteries. You require just (3) things which I'll describe in turn: POE Splitter, Outdoor Ethernet Cable and of course a POE Switch
POE Hat or POE Splitter: Since the POE Hat comes with a fan- which I hear come on quite frequently when used with my POE Pi-Cams- this probably wouldn't be suitable for this use-case. So I'll talk about the Splitter used to provide POE capabilities to your Pi only.
£9.90 currently from amazon.co.uk found here. It works by splitting the data from the power and diverting the Power from the POE Ethernet cable to the MicroUSB cable sticking out of the splitter which you connect to the Pi's power port. The Ethernet cable on the splitter of course connects to the Pi's Ethernet port. No tools requires and no instructions required for configuration.
Outdoor Ethernet Cable: Now you require an Ethernet cable to connect to your POE Splitter connected to the Pi in the birdbox to the POE Switch in your house (which I'll cover next). You CANNOT use a standard Ethernet cable- the weather will hammer it. You need one designed for use outdoors. The 20 meter cable (shown below) I use to power a wireless AP in my backyard for Internet surfing from my hammock can be bought here from amazon.co.uk for £11.39. Note that this model is sold in different lengths. It's 24AWG, so has thicker copper medium making it more suitable for POE purposes. And it's even Cat6 better still.
POE Switch: Now you need a POE Switch to connect the other end of that outdoor Ethernet cable to to power the Pi in the birdbox. I use a ZyXEL GS1900-8HP-GB0102F to power my POE Pi's- some using the POE Splitter, others using the POE Hat. In addition to a web interface, it can be managed at the CLI via SSH, has SNMP and IPv6 support. Has a very small form-factor and is fanless so very quiet. You can start the Pi even from a down state- ie you shut the Pi in the birdbox down using
sudo systemctl poweroff by accident. Log into the switch disable and re-enable the port and your pi is back to life. Can but it from amazon.co.uk (currently £88.35 ex-VAT) here
Hope this helps you with your project-