I planted a common lilac shrub, about 2 feet tall, in the spring.
. . .
There is healthy new growth, just not much of it. Overall, it looks
healthy, just small.

Don't know lilacs in particular, but generally for shrubs (and I guess
perennials too) this is pretty much what you'd hope for in the first 3
months: survival, some signs of growth/thriving. The slogan is
"sleep, creep, leap" (first year, plant doesn't do much, second year,
plant grows some, third year, plant is taking over your yard).