Will a watch band from Series 7/8 fit Series 5?

You can match most bands with any Apple Watch Series 3 or newer case of the same size. The 41mm bands work with 38mm and 40mm cases; the 45mm bands work with 42mm, 44mm and 49mm cases.

The Solo Loop and Braided Solo Loop bands are only compatible with Apple Watch SE and Apple Watch Series 4 or newer. The 40mm and 41mm cases work with band sizes 1–9; the 44mm and 45mm cases works with band sizes 1–12.

For more info, see the compatibility detail links in:

Buy Apple Watch Bands - Apple

If the case sizes are the same, then yes the watch band should fit your Series 5.