Kinda-sorta.
The wheel size has to match the frame/fork size, and the frame size should match the body size and riding type of the rider.
Overall smaller wheels will be stronger and easier to bring up to speed than bigger wheels. But bigger wheels roll smoother and offer a higher available gearing - which rarely is important outside a race setting.
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Yes, no, maybe. Unclear question. Same size front and rear - probably yes. Same size as every other fixie - no. Same size as found on regular bikes - sure. Pretty much any hub can be laced to any rim, and it's the rear hub that has to be a bit special to make a fixie - so you can have a fixie wheel in just about any size. Most common size is probably 700C, or 622 mm.