When visiting the Caribbean Islands, we often just pick up a local cabbie to give us a tour of the area. The tour generally lasts a couple of hours. Is it possible to pick up a local cab to sightsee in the Azores & Canary Islands in a similar manner?

One hint -- don't know how much activity there is but even though they are a territory of Spain, Fodors stashes Canary Islands threads on the Africa/Middle East forum.

Which Canary Islands?

I have never understood why Fodor's lists them under Africa when they are as much a part of Spain as Madrid or Barcelona. The Azores don't get listed as American even though they are on the American plate.

Basically yes

Some of the smaller Azores islands, will have airports but possibly no taxi. But for the main ones I'm sure the taxi at the airport would be happy, or you can book from your hotel/ship

The island chains are very different with the Canaries being very tourist focused while the Azores are a bit more laid back, though the constant visiting of ships is having an affect on local culture.

Azores are in the boundary zone between three plates so not naturally one or the other. ;-)

It depends on which Canary Islands Bilbo. Yes the eastern islands are more touristy but the smaller western islands are not.
I am assuming that OP is arriving on a ship, not by air - you wouldn't fly into any of these islands just for a few hours. If you are intending to stay on an island, or even if you aren't you could rent a car - it will be cheaper than a taxi, and you can see what you want to see. Certainly in the Canaries car rental is really cheap.

"Is it possible to pick up a local cab to sightsee in the Azores "
Yes, this is what we did a fex years ago. We rented a taxi for the day in Sao Miguel and for half a day in Terceira and Pico. We rented a car in San Jorge.

I have always seen Canary Is. posts tagged and listed under Spain and the Azores under Portugal, And, yes, you can pick up taxis to take you on a tour of all these islands. But, we rent cars, even for a day.

Yes.. . We are arriving by ship so have limited time on the islands.

Bedar, do you rent a car instead of taking a taxi due to the cost difference or because you want to go where you want to go? We're stopping in Lanzarote, Tenerife & Ponta Delgada & don't have any idea where we want to go on any of these islands. . . .hence we figure a taxi is our best bet.

big islands then.

Ponta Delgada; the two big lakes on the west end and Furnas towards the east are the two big drawers

OK, so you'll be coming off a ship. In that case I would use taxis.