Is it safe to replace a 4400mAh laptop battery with a 8800mAh battery

OK. After some digging, a typical mobile PC (from 2006) can handle 18V to 7.5V at the input either from an AC Adapter or a Battery. Also, the typical voltage for a battery is 11.1V or 14.8V depending upon the numbers of cells. The risk that 14.8V won't work on your laptop is extremely small.

Your laptop supports:

• 71 Wh, 4800 mAh (8-cell) Li-ion battery pack
• 44.4 Wh, 4000 mAh (6-cell) Li-ion battery
• 29.6 Wh, 2000 mAH (4-cell) Li-ion battery

Typically, laptops can work over a pretty wide range of input voltages. 14.8V doesn't sound too high to me. I can't find any specs from their website to confirm that 14.8V is OK, though.

Going from 6 to 8 cells will increase the voltage proportionally. Think of going from 6 to 8 AA batteries.

Here is an article on Notebook Battery Cells that presents an 8-cell battery being 14.4V as pretty common.

It sounds like you are talking about the kind of thing that has USB power ports for charging cell phones and tablets. If that is the case, then no, you can't power a laptop with it. What you want is a UPS.

Also their power rating is usually expressed in terms of a single cell, so 16 Ah at 4 volts = 64 Wh. Laptop batteries normally tell the capacity of an individual cell and how many cells they have, so Your 6 cell 4400 mAh laptop battery holds 105.6 Wh.

A typical UPS has a 7 or 8 Ah 12V battery so it holds 84 Wh. Higher end ones have dual batteries for twice the capacity, and output mains power for your laptop charger to convert to whatever the laptop accepts.