If an attacking unit with shielding gains a shield, keeps it after the engagement, then engages on defense the following turn, does it get a second shield?

The only effect that allows a unit to have more than 1 shield on it at any given time is Our Most Desperate Hour (seen below) and that only works on Character units.

Note that you must use Our Most Desperate Hour to gain the second shield after the first shield was already placed on that unit.

Page 23 of the rulebook

Shield Tokens

Some card effects place a shield token on units
or objectives. A shield token may not be assigned
to a card that already has a shield token.

Also, page 25 of the rulebook:

Shielding

When a card with the “Shielding” keyword is declared
as either attacker or defender, its controller may
immediately place one shield token on any friendly
unshielded participating unit, or to the engaged objective
card, if unshielded.

As mentioned, Our Most Desperate Hour is currently the only way to get a second shield on a card.