Yes, with strict limits
A green anole can be handled, but it belongs near the bottom of the list of lizards that tolerate it well. An adult is only a few inches of body on delicate bones, fast enough to vanish behind furniture in a blink, and wired to treat anything descending from above as a bird or snake. Handling is therefore an occasional privilege the lizard grants, not a daily part of its care. A realistic ceiling for a settled, confidently feeding anole is a few short sessions a week of a couple of minutes each, and many individuals are happier with none beyond essential enclosure work.
Give it time to settle first
Skip handling entirely for the first week or two in a new home. A freshly purchased or rehomed anole needs to establish its basking branch, learn where water droplets collect, and start hunting small insects reliably before any hand enters its world as more than a delivery tool. Brown colouration is a useful, if rough, stress signal in this species, since anoles shift between green and brown with temperature, mood, and surroundings. An anole that stays brown, hides constantly, and ignores food is telling you handling would be one pressure too many. Wait until it perches in the open, displays a relaxed green, and eats in your presence.
Technique for a lizard built to escape
Never reach down and close your fingers around an anole. Slide a hand slowly in at its own level and let it step or hop onto your palm, then raise the second hand as a walkway so it can move hand over hand under its own power. Work seated, close to the enclosure or over a soft surface, with doors shut and other pets out of the room, because a fall or a floor escape is the likeliest accident. Above all, never catch or lift it by the tail. Anoles can shed the tail as an escape trick, and while it regrows, the replacement is shorter and duller and the loss costs the lizard stored energy it cannot spare.
Reading when to put it back
End the session at the first persistent escape attempts, rapid colour darkening, gaping, or frantic scrambling. Brief curiosity, walking calmly and looking around with head raised, is the behaviour you want, and stopping while it lasts teaches the anole that hands are survivable. Chasing it around the enclosure to start a session cancels that lesson, so if it flees your approach twice, close the door and try another day. Offering a small insect from tongs after a calm session builds a food association without the risks of hand-catching prey together.
A display animal with occasional contact
The honest pleasure of a green anole is watching it behave: males extending a pink throat fan, rivals bobbing, a hunter stalking a fruit fly through the leaves. Handling adds little to that and can subtract a lot if overdone. Keep the balance right by making sure KinsLe Care Teams spreads this anole's daily misting, feeding, and light checks among assigned helpers, so its human contact is mostly calm care through the glass rather than repeated attempts to hold it.