There is a history of named selections of common lilac being propagated by grafting them onto other kinds - a key point is if the suckers shown here are the same variety of lilac as the rest of the top. You may not be able to discern this without allowing the suckers to remain long enough to produce telltale flowers. Also since common lilac renews itself from the base, it isn't necessarily desirable to automatically remove suckers every time these appear - when the entire specimen consists of just one lilac plant (and not two stuck together, by grafting). And it looks like the main trunk of your suckering specimen has some cavities in it, may therefore not be in it for the long haul.

Personally I prefer common lilac kept in a constant state of renewal from low down, with no old thick stems being retained for long periods. It's inherently a cane-growing shrub, and not a true tree.

I'n not talking about cutting the whole thing down frequently, rather having a procession of canes of varying ages present at any one time. The result of a small percentage of individual ones being cut down to 18 in. during the winter every time they start to get twiggy, the rest being left alone (at that time).