This is so cool! I do something similar for music - I have posters up for albums, and a small NFC tag is embedded behind the poster, so tapping my phone to the poster’s corner and clicking the notification begins playing it immediately. I really want to make a collection of cassette-or-record-style plastic cards with album art and NFC tags, and hook it up to a speaker, so I can place my album on a pedestal and it starts playing immediately.
I do the same for my CDs. I have them ripped on Plexamp and I place the NFC tags inside a corner of the booklet. So I tap the phone to the corner of a CD case instead. But my plan is to build a device like OP's so I don't even need the phone.
I use "NFC Tools" to write the URL, which I copy the album/song URL from Spotify and embed it. When I hold my phone near it, the URL appears in the notifications, and clicking it opens it in Spotify, which then starts playing it. I have a bag of 100 tags I got from a friend so no shortage there.
The regular https://open.spotify.com URL is registered to be handled by the Spotify app on iOS devices, so no bouncing from Safari - it opens straight away in Spotify. Nothing to play on a specific speaker, though - but my homepod mini is nearby so a tap to my poster then a tap to my speaker starts my listening session.
I have NFC Tools Pro from these guys, https://www.wakdev.com/en/ (but affiliated); I'm assuming you're referring to them. They're on Google Play store too.
I'd love to do something similar for myself, but having never worked with NFC I see there are tons of different options.
Would you mind mentioning what tags you got?