No, with the popular sites like PS and FT, it's not possible to login to the same site from multiple PC's. You'll either get kicked from the first login, or the second login will be refused, I don't remember which. It actually happened to me by accident once when I had left the client open on one PC and I tried to login from the other (I play from either my desktop or laptop depending on the situation).
There are different approaches to multi-tabling, and some are more dependent on monitor space than others.
Some folks prefer stacking, which is making each table the exact same size and stacking them exactly on top of each other (so you only ever see one table). When action is required on a table, it pops to the top of the stack. With stacking, monitor size or resolution is a non-issue because each table can remain full sized so there is theoretically no limit to the number of tables.
Others prefer to cascade their tables, which means each table is stacked but slightly offset such that you can see a piece of each table. This also allows a lot of tables on a single monitor because they're mostly stacked, although at some point you could have so many that some tables would be completely hidden (just as they are when taking the stacked approach).
The third option, tiling, is the one I prefer. With tiling, all windows are fully exposed at all times, usually arranged in a grid or matrix pattern. Resolution is key to tiling because you can only fit as many tables as your resolution and eyesight allow. You can shrink tables down until they're VERY tiny but it's more difficult to read cards and bets, click buttons, fit your HUD, etc. The larger the monitor (actually the resolution, not necessarily the physical size) the better, but even the largest monitor will only allow so many tiled tables, so most serious tilers use multiple monitors and tile tables across them.
I'm running three LCD monitors, two 24" and one 25.5", each at 1920x1200 for a total resolution of 5760x1200. Not just for poker, mind you -- my career is in the computer field and I've usually got different things going on different monitors -- but I certainly take advantage of it when playing poker. I can tile tables across all three and keep the tables at pretty much the same size as I would a single table, because I only 4-6 table. If one day I add more tables, I've still got a lot of room to shrink the tables down some but still keep them a nice size.
So another dedicated monitor would be the best approach. In a pinch, however, it is possible to use special software like MaxiVista and a network connection to actually use a laptop as a 2nd monitor.